<Question>
Express the quotient $413_5 \div 2_5$ in base 5.
</Question>

<think>
<Plan_1>
Okay, let me try to figure out how to divide these numbers in base 5. So the problem is 413 base 5 divided by 2 base 5, and I need to express the quotient in base 5. Hmm, base conversion and division can be a bit tricky, especially when not used to it. Let me start by recalling how numbers are represented in different bases.

First, I know that in base 5, each digit represents a power of 5. So the rightmost digit is 5^0 (which is 1), the middle digit is 5^1 (which is 5), and the leftmost digit is 5^2 (which is 25). So the number 413_5 would translate to 4*25 + 1*5 + 3*1 in decimal. Let me compute that to make sure I understand the value.
4*25 is 100, 1*5 is 5, and 3*1 is 3. Adding those up gives 100 + 5 + 3 = 108. So 413_5 is 108 in decimal.

Then, the divisor is 2_5, which is 2 in decimal. So if I were to do this division in decimal, 108 divided by 2 is 54. Then I need to convert 54 back to base 5. But maybe I should try doing the division directly in base 5 without converting to decimal? The problem might want practice in base 5 arithmetic.

Let me try both methods to cross-check, but perhaps focus on base 5 division.

Starting with base 5 division. The dividend is 413_5, which is 4 1 3 in base 5. The divisor is 2_5. So we need to divide 413_5 by 2_5.

Let me write it out like long division:

<prompt_1.1> Set up the long division of 413_5 by 2_5. </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
_5) 4 1 3
\end{execution_1.1>
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
First, we look at the leftmost digit of the dividend, which is 4. But in base 5, 4 divided by 2 is... well, 2 times 2 is 4, so that's 2 in base 5. So the first digit of the quotient is 2.

Then, multiply 2 (base 5) by 2 (base 5) which is 4. Subtract that from 4, which gives 0.

<prompt_2.1> Perform the division step by step as described in the above plan. </prompt_2.1>
</Plan_2>

<Execution_2>
<execution_2.1>
Bring down the next digit, which is 1. Now we have 0 1. Wait, in base 5, how does that work? So we have 1 in base 5. Now, 1 divided by 2 in base 5. Since 2 is larger than 1, we can't divide, so we put a 0 in the quotient here? But wait, maybe we need to consider it in decimal. Hmm, 1 divided by 2 is 0 with a remainder of 1. But in base 5, how is this handled? Maybe we proceed by bringing down the next digit.
</execution_2.1>
</Execution_2>

<Plan_3>
Based on execution_2.1:
Wait, the next digit is 3, so bringing down the 3 would make it 13 in base 5. Wait, 13 in base 5 is 1*5 + 3 = 8 in decimal. So 8 divided by 2 is 4. So the next digit in the quotient is 4.

So putting it all together, the quotient would be 2 0 4 in base 5, which is 204_5.

But let me verify this.

<prompt_3.1> Verify the result of the division by converting the quotient back to decimal. </prompt_3.1>

Alternatively, if I had converted 413_5 to decimal (108), divided by 2 (54), then converted 54 back to base 5. Let's do that to confirm.

<prompt_3.2> Verify the result by converting the numbers to decimal and performing the division. </prompt_3.2>
</Plan_3>

<Execution_3>
<execution_3.1>
If I multiply 204_5 by 2_5, does it give 413_5? Let's check. 

First, convert 204_5 to decimal: 2*25 + 0*5 + 4*1 = 50 + 0 + 4 = 54. Then 54 divided by 2 is 27. Now, convert 27 back to base 5: 27 divided by 5 is 5 with remainder 2. 5 divided by 5 is 1 with remainder 0. 1 divided by 5 is 0 with remainder 1. So writing the remainders from last to first: 1 0 2. Wait, that's 102_5. But 102_5 is 1*25 + 0*5 + 2*1 = 27, which is correct. But wait, 204_5 times 2_5 should be 413_5. But 54*2=108, which is 413_5. So 204_5 * 2_5 = 413_5.
</execution_3.1>

<execution_3.2>
54 divided by 5 is 10 with a remainder of 4. Then 10 divided by 5 is 2 with a remainder of 0. Then 2 divided by 5 is 0 with a remainder of 2. So writing the remainders from last to first, it's 2 0 4, so 204_5.
</execution_3.2>
</Execution_3>

<Plan_4>
Based on execution_3.1, execution_3.2:
But let me try doing the multiplication in base 5 to verify. 2_5 times 204_5.

<prompt_4.1> Verify the result by performing the multiplication in base 5. </prompt_4.1>

Based on execution_3.1:
Alternatively, maybe I can check by adding 204_5 + 204_5 in base 5. Let's see:

<prompt_4.2> Verify the result by performing addition in base 5. </prompt_4.2>
</Plan_4>

<Execution_4>
<execution_4.1>
Let's do it step by step. Starting from the rightmost digit:

4_5 times 2_5: 8 in decimal. 8 divided by 5 is 1 with remainder 3. So write down 3 and carry over 1.

Next digit: 0_5 times 2_5 is 0, plus the carryover 1 is 1. 1 divided by 2_5 is 0 with remainder 1.

Wait, this is getting confusing. Maybe it's better to perform the multiplication directly in base 5.

Multiplying 2_5 by 204_5:

Starting from the rightmost digit of 204_5:

2_5 * 4_5: In base 5, 2*4=8. But 8 divided by 5 is 1 with remainder 3. So we write down 3 and carry over 1.

Next digit: 2_5 * 0_5 = 0, plus carryover 1 gives 1. So write down 1. No carryover here.

Next digit: 2_5 * 2_5 = 4, which is less than 5, so write down 4.

So the result is 4 1 3 in base 5, which is 413_5.
</execution_4.1>

<execution_4.2>
Starting from the rightmost digit: 4 + 4. In base 5, 4 + 4 is 8 in decimal, which is 1*5 + 3, so write down 3 and carry over 1. Next middle digit: 0 + 0 plus the carryover 1 is 1. Next leftmost digit: 2 + 2 = 4. So the sum is 4 1 3, which is 413_5.
</execution_4.2>
</Execution_4>

<Plan_5>
Based on execution_4.1, execution_4.2:
Therefore, the quotient is 204_5. 

But just to make sure I didn't skip any steps in the division process, let me go through the division again in base 5 without converting to decimal.

<prompt_5.1> Recheck the division process in base 5 step by step. </prompt_5.1>
</Plan_5>

<Execution_5>
<execution_5.1>
Dividing 413_5 by 2_5:

Start with the leftmost digit: 4 divided by 2 is 2 in base 5. 2 times 2 is 4, subtract, remainder 0. Bring down the next digit 1. Since 1 is less than 2, we write 0 and carry down the next digit 3, making it 13 in base 5. 13 base 5 is 8 decimal. 8 divided by 2 is 4, so write 4. Multiply 4 by 2 (base 5) which is 8 decimal, which is 13 in base 5. Subtract 13 from 13, get 0. So quotient is 204_5.
</execution_5.1>
</Execution_5>

<Final_answer>
Therefore, the final answer is $\boxed{204_5}$.
</Final_answer>
</think>