
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
For a positive integer  $n>1$ , let  $g(n)$  denote the largest positive proper divisor of  $n$  and  $f(n)=n-g(n)$ . For example,  $g(10)=5, f(10)=5$  and  $g(13)=1,f(13)=12$ . Let  $N$  be the smallest positive integer such that  $f(f(f(N)))=97$ . Find the largest integer not exceeding  $\sqrt{N}$ 

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