
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
Given a $5 \times 5$ grid where the number in the $i$-th row and $j$-th column is denoted by \( a_{ij} \) (where \( a_{ij} \in \{0, 1\} \)), with the condition that \( a_{ij} = a_{ji} \) for \( 1 \leq i, j \leq 5 \). Calculate the total number of ways to fill the grid such that there are exactly five 1's in the grid.

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