
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 \( M \) be a \( 2014 \times 2014 \) invertible matrix, and let \( \mathcal{F}(M) \) denote the set of matrices whose rows are a permutation of the rows of \( M \). Find the number of matrices \( F \in \mathcal{F}(M) \) such that \( \operatorname{det}(M+F) \neq 0 \).

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