<Question>
Express $555_{10}$ in base $5$.
</Question>

<think>
<Plan_1>
Okay, so I need to convert the decimal number 555 into base 5. Hmm, how do I do that? Let me remember. I think the standard method is to divide the number by 5 repeatedly and record the remainders. Then, the base 5 number is the sequence of those remainders read in reverse order. Let me try to work through this step by step.

First, let me start with 555 divided by 5.

<prompt_1.1> Execute the division of 555 by 5 and record the quotient and remainder. </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
So, 555 ÷ 5. Well, 5 times 111 is 555, right? So the quotient is 111 and the remainder is 0. Okay, so the first remainder (which will be the least significant digit) is 0.
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Now take the quotient 111 and divide by 5 again.

<prompt_2.1> Execute the division of 111 by 5 and record the quotient and remainder. </prompt_2.1>
-----

Let me verify with another method. Let's compute the value of each digit in base 5. The digits in 4300_5 from right to left are positions 0, 1, 2, 3.

<prompt_2.2> Verify the conversion by computing the value of each digit in base 5. </prompt_2.2>
-----

Let me check once more to be sure.

Alternatively, maybe I can use another approach. Let's subtract the largest power of 5 less than 555 and see how many times each power fits.

<prompt_2.3> Use an alternative method to verify the conversion of 555 to base 5. </prompt_2.3>
</Plan_2>

<Execution_2>
<execution_2.1>
111 ÷ 5. Hmm, 5 times 22 is 110, so the quotient is 22 and the remainder is 1. So the next remainder is 1.
</execution_2.1>

<execution_2.2>
So:

4 * 5^3 = 4 * 125 = 500

3 * 5^2 = 3 * 25 = 75

0 * 5^1 = 0 * 5 = 0

0 * 5^0 = 0 * 1 = 0

Adding those up: 500 + 75 = 575, but wait, that's already over 555. Wait, that can't be right. Wait, no, 4*125 is 500, 3*25 is 75, so 500 +75 is 575, plus the zeros. But 575 is more than 555. Hmm, that suggests that maybe there's a mistake here.
</execution_2.2>

<execution_2.3>
The powers of 5 are: 5^0=1, 5^1=5, 5^2=25, 5^3=125, 5^4=625. Since 625 is larger than 555, the highest power we need is 5^3=125.

How many times does 125 go into 555? 555 ÷ 125 = 4.44, so 4 times. 4*125=500. Subtract that from 555: 555-500=55. So that's 4 in the 5^3 place.

Next, 5^2=25. How many times does 25 go into 55? 55 ÷25=2.2, so 2 times. 2*25=50. Subtract from 55: 55-50=5. So that's 2 in the 5^2 place.

Next, 5^1=5. How many times does 5 go into 5? 1 time. 1*5=5. Subtract from 5: 5-5=0. So that's 1 in the 5^1 place.

Finally, 5^0=1. How many times does 1 go into 0? 0 times. So that's 0 in the 5^0 place.

Putting those together: 4*125 + 2*25 +1*5 +0*1, which gives the digits 4,2,1,0. So 4210_5. Same result.
</execution_2.3>
</Execution_2>

<Plan_3>
Based on execution_2.1:
Take 22 ÷ 5.

<prompt_3.1> Execute the division of 22 by 5 and record the quotient and remainder. </prompt_3.1>
-----

Based on execution_2.2:
Wait, but 4*125 +3*25 is 500 +75=575. That's 10 more than 555. Hmm, so where did I go wrong here?

<prompt_3.2> Identify and correct the mistake in the alternative verification method. </prompt_3.2>
</Plan_3>

<Execution_3>
<execution_3.1>
22 ÷ 5 is 4 with a remainder of 2, right? Because 5*4=20, so remainder 2. So now the remainder is 2.
</execution_3.1>

<execution_3.2>
Wait, maybe I made a mistake in the conversion process. Let me check again.

Wait, when I converted 555 to base 5, I got 4210. Let's verify that again step by step.

First division: 555 ÷5=111, remainder 0.

Second: 111 ÷5=22, remainder 1.

Third: 22 ÷5=4, remainder 2.

Fourth: 4 ÷5=0, remainder 4.

So writing the remainders in reverse: 4,2,1,0. So 4210_5. Then converting back:

4*125=500, 2*25=50, 1*5=5, 0*1=0. 500+50=550, 550+5=555.
</execution_3.2>
</Execution_3>

<Plan_4>
Based on execution_3.1:
Then take 4 ÷ 5.
That's 0 with a remainder of 4. So the remainder here is 4.

Now, since the quotient is 0, we stop. The remainders we got were 0, 1, 2, 4. Reading them in reverse order gives 4, 2, 1, 0. So the base 5 number is 4210_5.
-----

Based on execution_3.2:
Ah, there was a miscalculation in my alternative verification. So the correct conversion is indeed 4210_5. Therefore, the initial conversion was correct.
</Plan_4>

<Execution_4>
No prompt to execute.
</Execution_4>

<Final_answer>
Therefore, $555_{10}$ in base $5$ is $\boxed{4210_5}$.
</Final_answer>
</think>