<Question>
What is the base six equivalent of $999_{10}$?
</Question>

<think>
<Plan_1>
Okay, so I need to convert the decimal number 999 into base six. Hmm, base six, right? Let me remember how to do this. I think the general method is to divide the number by 6 repeatedly and record the remainders each time. Then, the base six number is the sequence of those remainders read in reverse order. Let me try to work through this step by step.

First, let me confirm the process. To convert a decimal number to another base, you divide the number by the base, write down the remainder, then repeat the process with the quotient until the quotient is zero. Then, the number in the new base is the remainders read from last to first.

So, starting with 999 divided by 6.

<prompt_1.1> Execute the conversion of the decimal number 999 into base six using the division and remainder method. </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
Let me do that. 999 ÷ 6. Hmm, 6 times 166 is 996, right? So 6*166=996. Then 999 minus 996 is 3. So the first remainder is 3. The quotient is 166.
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Now, take 166 and divide by 6.

<prompt_2.1> Continue the conversion process with the new quotient 166. </prompt_2.1>
</Plan_2>

<Execution_2>
<execution_2.1>
Let's see, 6*27 is 162. So 166 minus 162 is 4. So the remainder is 4. The quotient is now 27.
</execution_2.1>
</Execution_2>

<Plan_3>
Based on execution_2.1:
Next, take 27 and divide by 6.

<prompt_3.1> Continue the conversion process with the new quotient 27. </prompt_3.1>
</Plan_3>

<Execution_3>
<execution_3.1>
6*4 is 24, so remainder is 3. Quotient is 4.
</execution_3.1>
</Execution_3>

<Plan_4>
Based on execution_3.1:
Now take 4 and divide by 6.

<prompt_4.1> Continue the conversion process with the new quotient 4. </prompt_4.1>
</Plan_4>

<Execution_4>
<execution_4.1>
6*0 is 0, so the remainder is 4. The quotient is 0. So we stop here.
</execution_4.1>
</Execution_4>

<Plan_5>
Based on execution_4.1:
Now, the remainders we got were 3, 4, 3, 4. But since we read them in reverse order of how we got them, right?

<prompt_5.1> Read the remainders in reverse order to form the base six number. </prompt_5.1>
</Plan_5>

<Execution_5>
<execution_5.1>
The first remainder was 3 (from the first division), then 4, then 3, then 4. But when we reverse that order, it becomes 4, 3, 4, 3. So the base six number should be 4343.
</execution_5.1>
</Execution_5>

<Plan_6>
Based on execution_5.1:
Wait, let me check that.

Let me verify by converting 4343 base six back to decimal to see if it equals 999.

<prompt_6.1> Verify the conversion by converting 4343 base six back to decimal. </prompt_6.1>

But let me make sure I didn't make any mistakes in the conversion steps. Let me go through the division again.

<prompt_6.2> Recheck the conversion steps to ensure accuracy. </prompt_6.2>

Alternatively, maybe there's another way to check. Let me see. Let's compute the powers of 6 to see how the digits correspond.

<prompt_6.3> Use powers of 6 to verify the conversion result. </prompt_6.3>

But just to make sure, maybe I can try another method. Let's see, maybe using the expansion method.

<prompt_6.4> Try another method to confirm the conversion result. </prompt_6.4>
</Plan_6>

<Execution_6>
<execution_6.1>
Starting from the right, the digits are 4, 3, 4, 3. So, positions 0 to 3:

4*6^3 + 3*6^2 + 4*6^1 + 3*6^0.

Calculating each term:

4*216 = 864

3*36 = 108

4*6 = 24

3*1 = 3

Adding them up: 864 + 108 = 972; 972 +24=996; 996+3=999.
</execution_6.1>

<execution_6.2>
Starting with 999:

999 ÷ 6 = 166 remainder 3

166 ÷ 6 = 27 remainder 4

27 ÷ 6 = 4 remainder 3

4 ÷ 6 = 0 remainder 4

So the remainders from last to first are 4, 3, 4, 3, hence 4343.
</execution_6.2>

<execution_6.3>
6^0 = 1

6^1 = 6

6^2 = 36

6^3 = 216

6^4 = 1296

Since 6^4 is 1296, which is larger than 999, the highest power we need is 6^3=216.

Now, how many times does 216 go into 999? 999 ÷ 216. Let's calculate:

216*4 = 864

216*5 = 1080, which is too much. So 4 times. 4*216=864. Subtract that from 999: 999 - 864 = 135.

Now, the next lower power is 6^2=36. How many times does 36 go into 135? 135 ÷ 36 = 3.75, so 3 times. 3*36=108. Subtract from 135: 135 - 108 = 27.

Next power is 6^1=6. How many times does 6 go into 27? 4 times (since 6*4=24). Subtract 24 from 27: 3.

Finally, 6^0=1. How many times does 1 go into 3? 3 times. So the coefficients are 4 (for 6^3), 3 (for 6^2), 4 (for 6^1), and 3 (for 6^0). So that gives 4343 base six. Which matches what I got earlier. So that's reassuring.
</execution_6.3>

<execution_6.4>
For base six, the places are 6^3, 6^2, 6^1, 6^0. Let's see what the highest power of 6 less than 999 is.

6^0 = 1

6^1 = 6

6^2 = 36

6^3 = 216

6^4 = 1296

Oh, 6^4 is 1296, which is more than 999, so the highest power is 6^3 = 216.

So how many times does 216 go into 999? 999 ÷ 216. Let's calculate that.

216*4 = 864

216*5 = 1080, which is too much. So 4 times. So the coefficient for 6^3 is 4. Then subtract 864 from 999: 999 - 864 = 135.

Now, move to the next lower power, which is 6^2 = 36.

How many times does 36 go into 135?

36*3 = 108

36*4 = 144, which is too much. So 3 times. Coefficient is 3. Subtract 108 from 135: 135 - 108 = 27.

Next power is 6^1 = 6.

How many times does 6 go into 27? 6*4=24, 6*5=30 which is too much. So 4 times. Coefficient is 4. Subtract 24 from 27: 27 -24=3.

Finally, the last digit is the remainder, which is 3. So putting it all together, the coefficients are 4 (for 6^3), 3 (for 6^2), 4 (for 6^1), and 3 (for 6^0). So 4343 base six. Which matches the previous result. So that's good.
</execution_6.4>
</Execution_6>

<Plan_7>
Based on execution_6.1, execution_6.2, execution_6.3, execution_6.4:
I think that's thorough.
I checked it two different ways: dividing repeatedly by 6 and getting the remainders, and also by expanding the number into powers of 6. Both methods gave me the same result, and verifying by converting back to decimal also worked. So I feel confident that 999 in base 10 is 4343 in base 6.
</Plan_7>

<Execution_7>
No prompt to execute.
</Execution_7>

<Final_answer>
Therefore, the base six equivalent of $999_{10}$ is $\boxed{4343_6}$.
</Final_answer>
</think>