
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
Define a function $f_1$ on the positive integers by setting $f_1(1)=1$ and for $n>1$ whose prime factorization is $n=p_1^{e_1}p_2^{e_2}\cdots p_k^{e_k}$, let:
\[f_1(n)=(2p_1+1)^{e_1}(2p_2+1)^{e_2}\cdots (2p_k+1)^{e_k}.\]
Then, for every $m\ge 2$, define $f_m(n)=f_1(f_{m-1}(n))$. Determine how many values of $N$ in the range $1 \le N \le 450$ result in an unbounded sequence $(f_1(N), f_2(N), f_3(N), \dots)$.
**A)** 19
**B)** 20
**C)** 21
**D)** 22
**E)** 23

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