
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
In every cell of a \(5 \times 5\) square, there is a number. The numbers in any given row (any row) and any given column (any column) form an arithmetic progression. The numbers in the corners of the square are \(1, 25, 17, 81\). What number is in the center cell? Do not forget to justify your answer.

\[
\begin{array}{|c|c|c|c|c|}
\hline
1 &  &  &  & 25 \\
\hline
 &  &  &  &  \\
\hline
 &  & x &  &  \\
\hline
 &  &  &  &  \\
\hline
81 &  &  &  & 17 \\
\hline
\end{array}
\]

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