
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 \( g_1 \) defined on the positive integers by \( g_1(1) = 1 \) and \( g_1(n) = (p_1-1)^{e_1-1} (p_2-1)^{e_2-1} \cdots (p_k-1)^{e_k-1} \), where \( n = p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k} \) is the prime factorization of \( n > 1 \), and for every \( m \geq 2 \), let \( g_m(n) = g_1(g_{m-1}(n)) \), determine the number of integers \( N \) in the range \( 1 \leq N \leq 100 \) for which the sequence \( (g_1(N), g_2(N), g_3(N), \dots) \) is unbounded.

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