<Question>
The sum of 27 consecutive positive integers is $3^7$. What is their median?
</Question>

<think>
<Plan_1>
Okay, so I need to find the median of 27 consecutive positive integers whose sum is 3^7. Let me start by recalling some basic concepts. 

First, the median of a set of numbers is the middle number when they are arranged in order. Since there are 27 numbers, which is an odd count, the median will be the 14th number in the ordered list. But I also remember that for consecutive numbers, the median is the same as the average of the sequence. That's because consecutive numbers form an arithmetic sequence, where the average is equal to the median. So maybe I don't have to find the entire sequence; perhaps I can just find the average, which would be the median.
Let me check that thought.

The sum of the numbers is given as 3^7. Let me compute what 3^7 is.

<prompt_1.1> Calculate the value of 3^7 as the sum of 27 consecutive integers. </prompt_1.1>

Since the average of the numbers is the sum divided by the count, which is 27. So average = 3^7 / 27. Let me compute that.

<prompt_1.2> Calculate the average and median of the 27 consecutive integers. </prompt_1.2>
</Plan_1>

<Execution_1>
<execution_1.1>
3^1 is 3, 3^2 is 9, 3^3 is 27, 3^4 is 81, 3^5 is 243, 3^6 is 729, and 3^7 is 2187. So the sum of these 27 consecutive numbers is 2187.
</execution_1.1>

<execution_1.2>
27 is 3^3, so 3^7 divided by 3^3 is 3^(7-3) = 3^4. 3^4 is 81. So the average is 81. Therefore, the median is 81.
</execution_1.2>
</Execution_1>

<Plan_2>
Based on execution_1.2:
Wait, that seems straightforward. But let me make sure I'm not missing anything here.

<prompt_2.1> Verify the calculation of the median using the properties of consecutive numbers. </prompt_2.1>

But let me verify this another way to be sure. Let's denote the first integer in the sequence as n. Then the sequence is n, n+1, n+2, ...., up to n+26 (since there are 27 terms). The sum of an arithmetic sequence is (number of terms)/2 * (first term + last term). So sum = 27/2 * (n + (n + 26)) = 27/2 * (2n + 26) = 27*(n + 13). According to the problem, this sum is 3^7 = 2187. So 27*(n + 13) = 2187. Dividing both sides by 27, we get n + 13 = 2187 / 27. Let me compute that.

<prompt_2.2> Solve for the first term of the sequence and verify the median using an alternative method. </prompt_2.2>
</Plan_2>

<Execution_2>
<execution_2.1>
The key here is that in an arithmetic sequence, the median is equal to the mean. Since these are consecutive integers, they form an arithmetic sequence with a common difference of 1. Therefore, the average (mean) of the sequence is equal to the median. Hence, if the average is 81, the median is 81.
</execution_2.1>

<execution_2.2>
2187 divided by 27. Let me do this division step by step. 27*80 = 2160. 2187 - 2160 = 27. So 27*81 = 2187. Therefore, n + 13 = 81. Therefore, n = 81 - 13 = 68. So the first number is 68. Then the sequence is 68, 69, ...., up to 68 + 26 = 94. The median is the 14th term. Since the sequence starts at 68, the 14th term is 68 + 13 = 81. So that confirms it. The median is indeed 81.
</execution_2.2>
</Execution_2>

<Plan_3>
Based on execution_2.2:
But let me just go through another check to make sure. Let's confirm that the sum of numbers from 68 to 94 is indeed 2187. The sum can be calculated as (number of terms)*(first term + last term)/2. So 27*(68 + 94)/2.

<prompt_3.1> Perform a final verification of the sum and conclude the solution. </prompt_3.1>
</Plan_3>

<Execution_3>
<execution_3.1>
68 + 94 is 162. 162 divided by 2 is 81. 81 multiplied by 27 is indeed 2187, which matches the given sum. Therefore, all checks out. 

So the median is 81. I don't see any gaps in this reasoning. Both methods of calculating the median—directly through the average and by finding the first term and then the 14th term—result in 81. Also, verifying the sum confirms that the sequence is correct.
</execution_3.1>
</Execution_3>

<Final_answer>
Therefore, the median of the $27$ consecutive positive integers is $\boxed{81}$.
</Final_answer>
</think>