
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

There are exactly 120 ways to color five cells in a $5 \times 5$ grid so that there is exactly one colored cell in each row and each column.

There are exactly 96 ways to color five cells in a $5 \times 5$ grid without the corner cell, such that there is exactly one colored cell in each row and each column.

How many ways are there to color five cells in a $5 \times 5$ grid without two corner cells so that there is exactly one colored cell in each row and each column?

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