
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 set \( A \) such that \( f(f(x)) \) is a constant function. Find the remainder when \( N \) is divided by 1000.

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