
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
Given the function $f_{1}$ defined by $f_{1}(1)=1$ and $f_{1}(n)=(p_{1}+1)^{e_{1}-1}(p_{2}+1)^{e_{2}-1}\cdots (p_{k}+1)^{e_{k}-1}$ for any prime factorization $n=p^{e_{1}}p_{2}^{e_{2}}\cdots p_{k}^{e_{k}}$ of $n>1$, and for every $m\ge 2$, $f_{m}(n)=f_{1}(f_{m-1}(n))$, calculate the number of integers $N$ in the range $1\le N\le 400$ for which the sequence $(f_{1}(N),f_{2}(N),f_{3}(N),\dots)$ is unbounded.

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