
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  $N$  be the number of functions  $f:\{1,2,3,4,5,6,7,8,9,10\} \rightarrow \{1,2,3,4,5\}$  that have the property that for  $1\leq x\leq 5$  it is true that  $f(f(x))=x$ . Given that  $N$  can be written in the form  $5^a\cdot b$  for positive integers  $a$  and  $b$  with  $b$  not divisible by  $5$ , find  $a+b$ . 

[i]Proposed by Nathan Ramesh

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