
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 for \( n=p_1^{e_1}p_2^{e_2}\cdots p_k^{e_k} \) (the prime factorization of \( n > 1 \)), set \[ g_1(n) = (p_1+2)^{e_1-1}(p_2+2)^{e_2-1}\cdots (p_k+2)^{e_k-1}. \]
For every \( m \geq 2 \), let \( g_m(n) = g_1(g_{m-1}(n)) \). Determine for how many integers \( N \) in the range \( 1 \leq N \leq 450 \) is the sequence \( (g_1(N), g_2(N), g_3(N), \dots) \) unbounded.
A) 18
B) 19
C) 20
D) 21
E) 22

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