
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 $f_1$ defined by $f_1(1)=2$ and $f_1(n)=(p_1+2)^{e_1-1}(p_2+2)^{e_2-1}$ for $n=p_1^{e_1}p_2^{e_2}$, where $p_1 < p_2$, and for every $m\ge 2$, $f_m(n)=f_1(f_{m-1}(n))$, calculate the number of integers $N$ in the range $1\le N\le 300$ that make the sequence $(f_1(N),f_2(N),f_3(N),\dots )$ ultimately become a power of $2$.

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