
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  $\mathbb{F}_{13} = {\overline{0}, \overline{1}, \cdots, \overline{12}}$  be the finite field with  $13$  elements (with sum and product modulus  $13$ ). Find how many matrix  $A$  of size  $5$  x  $5$  with entries in  $\mathbb{F}_{13}$  exist such that  $$ A^5 = I $$   where  $I$  is the identity matrix of order  $5$ 

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