
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
For each positive integer \( n \), define \( f_1(n) \) as three times the number of positive integer divisors of \( n \). For \( j \geq 2 \), let \( f_j(n) = f_1(f_{j-1}(n)) \). Determine how many values of \( n \leq 100 \) exist such that \( f_{100}(n) = 27 \).
A) 4
B) 5
C) 6
D) 7

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