
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system

Let \( c_{n}=11 \ldots 1 \) be a number in which the decimal representation contains \( n \) ones. Then \( c_{n+1}=10 \cdot c_{n}+1 \). Therefore:

\[ c_{n+1}^{2}=100 \cdot c_{n}^{2} + 22 \ldots 2 \cdot 10 + 1 \]

For example,

\( c_{2}^{2}=11^{2}=(10 \cdot 1+1)^{2}=100+2 \cdot 10+1=121 \),

\( c_{3}^{2} = 111^{2} = 100 \cdot 11^{2} + 220 + 1 = 12100 + 220 + 1 = 12321 \),

\( c_{4}^{2} = 1111^{2} = 100 \cdot 111^{2} + 2220 + 1 = 1232100 + 2220 + 1 = 1234321 \), etc.

We observe that in all listed numbers \( c_{2}^{2}, c_{3}^{2}, c_{4}^{2} \), the digit with respect to which these numbers are symmetric (2 in the case of \( c_{2}^{2}, 3 \) in the case of \( c_{3}^{2}, 4 \) in the case of \( c_{4}^{2} \)) coincides with the number of ones in the number that was squared. 

The given number \( c=123456787654321 \) is also symmetric with respect to the digit 8, which suggests that it might be the square of the number \( c_{8} = 11111111 \). This can be verified by performing multiplication by columns or using the recursive relation.

Present the answer in LaTex format: \boxed{Your answer}
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