### Component 1
<description>
Planning the overall approach to solve the problem.
</description>

<plan>
Okay, let me try to figure out this problem step by step. So, the problem is about reversing the digits of 2005 to get 5002, which is equal to a product of three distinct primes a, b, c. Then, we need to find how many other positive integers are products of exactly three distinct primes p1, p2, p3 such that the sum p1 + p2 + p3 equals the sum a + b + c. 

First, let's break down the problem into parts. The first task is to factorize 5002 into three distinct primes. Then, compute the sum of those primes. The second part is to find all other numbers (different from 5002) that are the product of exactly three distinct primes whose sum is equal to the sum we found. Finally, count how many such numbers there are.
</plan>

<prompt>
Plan the steps needed to solve the problem of finding other positive integers that are products of exactly three distinct primes with a specific sum.
</prompt>

<execution>
None
</execution>


### Component 2
<description>
Reversing the digits of 2005 to get 5002.
</description>

<plan>
Starting with the first part: reversing the digits of 2005 gives 5002. So, we need to factor 5002 into three distinct primes.
</plan>

<prompt>
Reverse the digits of 2005 to find the number 5002.
</prompt>

<execution>
None
</execution>


### Component 3
<description>
Factorizing 5002 into three distinct primes.
</description>

<plan>
Let me start by prime factorization of 5002.

5002 is an even number, so it's divisible by 2. Let's divide 5002 by 2:
</plan>

<prompt>
Factorize 5002 into its prime components.
</prompt>

<execution>
5002 ÷ 2 = 2501.

Now, 2501 is the remaining factor. Let's check if 2501 is a prime or can be factored further. To check if 2501 is prime, I can test divisibility by primes up to its square root. The square root of 2501 is approximately 50, so I need to check primes up to 50.

Testing divisibility by 3: 2 + 5 + 0 + 1 = 8, which is not divisible by 3. So, 2501 ÷ 3 is not an integer.

Next, 5: It doesn't end in 0 or 5, so not divisible by 5.

7: Let's divide 2501 by 7. 7*357 = 2499, which is 2501 - 2499 = 2. So, remainder 2. Not divisible by 7.

11: Using the rule for 11: (2 + 0) - (5 + 1) = (2) - (6) = -4, which isn't divisible by 11. So, not divisible by 11.

13: Let's try 13*192 = 2496. 2501 - 2496 = 5, so remainder 5. Not divisible by 13.

17: 17*147 = 2499. 2501 - 2499 = 2. Remainder 2. Not divisible by 17.

19: 19*131 = 2489. 2501 - 2489 = 12. Not divisible.

23: 23*108 = 2484. 2501 - 2484 = 17. Not divisible.

29: 29*86 = 2494. 2501 - 2494 = 7. Not divisible.

31: 31*80 = 2480. 2501 - 2480 = 21. 21 ÷ 31 isn't whole.

37: 37*67 = 2479. 2501 - 2479 = 22. Not divisible.

41: 41*61 = 2501. Wait, 41*60 = 2460, plus 41 is 2501. So, 41*61 = 2501. Let me check that multiplication:

41*60 = 2460, plus 41*1 = 41, so 2460 + 41 = 2501. Correct. Therefore, 2501 = 41*61.

So now, 5002 factors into 2 * 41 * 61. Are all these primes? Let's verify:

2 is prime. 41 is prime (divisible only by 1 and 41). 61 is also prime (divisible by 1 and 61). So, yes, 5002 is the product of three distinct primes: 2, 41, and 61.
</execution>


### Component 4
<description>
Calculating the sum of the prime factors of 5002.
</description>

<plan>
Therefore, a, b, c are 2, 41, 61. The sum a + b + c is 2 + 41 + 61 = 104.
</plan>

<prompt>
Calculate the sum of the prime factors of 5002.
</prompt>

<execution>
So, the sum we're looking at is 104.
</execution>


### Component 5
<description>
Planning the approach to find other numbers that are products of three distinct primes with the same sum.
</description>

<plan>
Now, the problem is asking: How many other positive integers (other than 5002) are the product of exactly three distinct primes p1, p2, p3 such that p1 + p2 + p3 = 104?

So, we need to find all sets of three distinct primes where their sum is 104, compute their product, and count how many unique products there are excluding 5002 itself.
</plan>

<prompt>
Plan how to find other numbers that are products of three distinct primes with the same sum as the prime factors of 5002.
</prompt>

<execution>
First, let's note that 2 is the only even prime. The sum of three primes is 104, which is an even number. If all three primes are odd, their sum would be odd + odd + odd = odd. But 104 is even. Therefore, there must be at least one even prime in the triplet, which can only be 2. So, one of the primes must be 2, and the other two must be odd primes adding up to 102 (since 104 - 2 = 102). So, the problem reduces to finding pairs of distinct primes (p, q) such that p + q = 102, and then the triplet is (2, p, q). Then, each such triplet gives a product 2*p*q, and we need to count how many such products exist, excluding 5002.

Wait, but we need to check if there are triplets where two primes are even? But since 2 is the only even prime, the other two primes must be odd. So, in this case, only one even prime is possible. Therefore, all triplets must include 2 and two primes that add to 102. Therefore, we need to find all pairs of distinct primes that add to 102.

So, the problem reduces to: find the number of unordered pairs (p, q) where p and q are distinct primes, p + q = 102. Each such pair gives a product 2*p*q. We need to count all such products, exclude 2*41*61=5002, and the count of remaining products is the answer.
</execution>


### Component 6
<description>
Planning the detailed steps to find pairs of primes that sum to 102.
</description>

<plan>
Therefore, first, find all prime pairs (p, q), p < q, such that p + q = 102.

Let me list all primes less than 102, and check which pairs add up to 102.

Alternatively, since p and q are primes with p + q = 102, then p must be a prime less than 51 (since p < q, so p < 102 - p => p < 51). Wait, if p < q, then p < 102/2 = 51. So p must be a prime less than 51, and q = 102 - p must also be prime.
</plan>

<prompt>
Plan the detailed steps to find pairs of primes that sum to 102.
</prompt>

<execution>
Therefore, we can iterate through primes p from 3 up to 47 (since next prime after 47 is 53, which is over 51), and check if 102 - p is prime.

Wait, but p can be 2? Wait, no. Because if p=2, then q=100, but 100 is not prime. Since 102 - 2 = 100, which is even and not prime. So, since we already established that one prime must be 2, but wait, no. Wait, the triplet must include 2, but in this case, when we split 104 as 2 + p + q, p and q are primes. Wait, actually, no. Wait, earlier logic was that the triplet must include 2 because otherwise, three odd primes would sum to an odd number, but 104 is even. Therefore, exactly one of the primes must be 2, and the other two sum to 102. Therefore, the other two primes must be odd and sum to 102.

Therefore, all triplets will be (2, p, q) where p and q are odd primes, p + q = 102, and p ≠ q. So, the pairs (p, q) with p < q, p + q = 102, both primes.

Therefore, we need to find all such pairs, compute 2*p*q for each pair, and count them, then subtract 1 (for 5002) to get the number of other positive integers.

So, the plan is:

1. Generate all primes less than 102.

2. For each prime p in 3 to 50 (since p < q, so p < 51), check if 102 - p is prime.

3. If yes, then (p, 102 - p) is a valid pair.

4. For each such pair, compute 2*p*(102 - p).

5. Collect all such products, remove 5002, and count the remaining.

Therefore, let's proceed step by step.
</execution>


### Component 7
<description>
Listing all primes up to 102.
</description>

<plan>
First, generate primes up to 102. Let me list primes below 102.
</plan>

<prompt>
List all prime numbers up to 102.
</prompt>

<execution>
Primes up to 102:

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101.

Now, we need to check for each prime p from 3 up to 47 (since 102 - 47 = 55, which is within the prime list? Wait, 55 is not prime. Wait, but primes go up to 101. Let me clarify.

Wait, if p is a prime less than 51 (since p < q), then q = 102 - p is greater than 51. So q can be a prime up to 102 - 3 = 99. But 99 is not prime. So we need q to be prime. So, primes p where p is a prime less than 51 and 102 - p is also prime.

Therefore, for each prime p in 3, 5, 7, ...., 47 (since 47 is the largest prime less than 51), check if 102 - p is prime.

Let me list primes less than 51:

3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47.

These are all primes below 51.
</execution>


### Component 8
<description>
Finding pairs of primes that sum to 102.
</description>

<plan>
Now, for each p in this list, compute q = 102 - p and check if q is prime.

Let's go through each:
</plan>

<prompt>
Find pairs of primes (p, q) such that p + q = 102.
</prompt>

<execution>
1. p = 3: q = 102 - 3 = 99. 99 is not prime (divisible by 9 and 11). Not valid.

2. p = 5: q = 102 - 5 = 97. 97 is prime. Valid pair (5, 97).

3. p = 7: q = 102 - 7 = 95. 95 = 5*19. Not prime. Not valid.

4. p = 11: q = 102 - 11 = 91. 91 = 7*13. Not prime. Invalid.

5. p = 13: q = 102 - 13 = 89. 89 is prime. Valid pair (13, 89).

6. p = 17: q = 102 - 17 = 85. 85 = 5*17. Not prime. Invalid.

7. p = 19: q = 102 - 19 = 83. 83 is prime. Valid pair (19, 83).

8. p = 23: q = 102 - 23 = 79. 79 is prime. Valid pair (23, 79).

9. p = 29: q = 102 - 29 = 73. 73 is prime. Valid pair (29, 73).

10. p = 31: q = 102 - 31 = 71. 71 is prime. Valid pair (31, 71).

11. p = 37: q = 102 - 37 = 65. 65 = 5*13. Not prime. Invalid.

12. p = 41: q = 102 - 41 = 61. 61 is prime. Valid pair (41, 61). But wait, this is the original triplet (2, 41, 61) which gives 5002. So this is the pair that leads to the original number.

13. p = 43: q = 102 - 43 = 59. 59 is prime. Valid pair (43, 59).

14. p = 47: q = 102 - 47 = 55. 55 = 5*11. Not prime. Invalid.

So, the valid pairs (p, q) where p and q are primes adding to 102 are:

(5, 97), (13, 89), (19, 83), (23, 79), (29, 73), (31, 71), (41, 61), (43, 59).
</execution>


### Component 9
<description>
Identifying the number of valid prime pairs.
</description>

<plan>
Wait, that's 8 pairs.
</plan>

<prompt>
Count the number of valid prime pairs found.
</prompt>

<execution>
However, note that (41, 61) is the pair used in 5002.
</execution>


### Component 10
<description>
Calculating the products of the valid prime pairs.
</description>

<plan>
So, the products for each pair are:

For each pair (p, q):
</plan>

<prompt>
Calculate the product 2*p*q for each valid prime pair (p, q).
</prompt>

<execution>
Product = 2 * p * q

So:

1. 2 * 5 * 97 = 2 * 485 = 970

2. 2 * 13 * 89 = 2 * 1157 = 2314

3. 2 * 19 * 83 = 2 * 1577 = 3154

4. 2 * 23 * 79 = 2 * 1817 = 3634

5. 2 * 29 * 73 = 2 * 2117 = 4234

6. 2 * 31 * 71 = 2 * 2201 = 4402

7. 2 * 41 * 61 = 5002 (the original number)

8. 2 * 43 * 59 = 2 * 2537 = 5074

Therefore, these products are:

970, 2314, 3154, 3634, 4234, 4402, 5002, 5074.

Now, we need to exclude 5002 since the problem asks for "other positive integers", so the count is 8 products in total minus 1 (excluding 5002) = 7.

But wait, before we conclude that, we need to make sure that all these products are distinct. Let's check if any of these products are the same. Let's list them:

970, 2314, 3154, 3634, 4234, 4402, 5002, 5074.

Looking at these numbers, they all seem distinct. No duplicates here. So, 8 products in total, subtract 1 for the original number, gives 7.
</execution>


### Component 11
<description>
Verifying the calculations of the products.
</description>

<plan>
But wait, let me confirm each calculation to be sure.

Calculations:
</plan>

<prompt>
Verify the calculations of the products 2*p*q for each valid prime pair.
</prompt>

<execution>
1. 2*5*97: 5*97=485, 485*2=970. Correct.

2. 2*13*89: 13*89=1157, 1157*2=2314. Correct.

3. 2*19*83: 19*83=1577, 1577*2=3154. Correct.

4. 2*23*79: 23*79=1817, 1817*2=3634. Correct.

5. 2*29*73: 29*73=2117, 2117*2=4234. Correct.

6. 2*31*71: 31*71=2201, 2201*2=4402. Correct.

7. 2*41*61=5002. Correct.

8. 2*43*59: 43*59=2537, 2537*2=5074. Correct.
</execution>


### Component 12
<description>
Concluding the solution by confirming the number of valid products.
</description>

<plan>
Therefore, the answer is 7.
</plan>

<prompt>
Conclude the solution by confirming the number of valid products excluding 5002.
</prompt>

<execution>
Hence, there are 7 other positive integers besides 5002 that are products of exactly three distinct primes whose sum is 104.

But wait, hold on. Let me check again if there's any other triplet that doesn't include 2. Wait, earlier reasoning said that since the sum of three primes is even (104), one of them must be 2. Because if all three primes are odd, their sum would be odd (since odd + odd + odd = odd). But 104 is even, so one of them must be even. The only even prime is 2. Therefore, all triplets must include 2, and the other two primes are odd and sum to 102. Therefore, the list above is complete. So, all possible triplets are accounted for, and their products are the 8 listed, with 5002 being one of them. Therefore, the remaining 7 are the answer.

But let me check again if there are primes p, q, r (all odd) that sum to 104. Wait, sum of three odd numbers is odd. 104 is even, so that's impossible. So, only triplets with exactly one even prime (which is 2) and two odd primes. Hence, the above list is complete. Therefore, the answer is 7.
</execution>


### Component 13
<description>
Presenting the final answer.
</description>

<plan>
**Final Answer**
</plan>

<prompt>
Present the final answer in a boxed format.
</prompt>

<execution>
\boxed{7}
</execution>
