
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 $A = \{1, 2, 3, 4, 5, 6, 7, 8\}$, and let $N$ be the number of functions $f$ from set $A$ to itself such that $f(f(x)) = c$ for all $x$ except for one specified $x = a \in A$, for which $f(f(a)) = b \neq c$. Find $N$.

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