
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 the function $g_1$ on the positive integers by setting $g_1(1)=1$ and if $n=q_1^{d_1}q_2^{d_2}\cdots q_k^{d_k}$ is the prime factorization of $n>1$, then \[g_1(n)=(q_1+2)^{d_1-1}(q_2+2)^{d_2-1}\cdots (q_k+2)^{d_k-1}.\]
For every $m\ge 2$, let $g_m(n)=g_1(g_{m-1}(n))$. For how many integers $N$ in the range $1\le N\le 500$ is the sequence $(g_1(N),g_2(N),g_3(N),\dots )$ potentially unbounded?
A) 47
B) 48
C) 49
D) 50

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