
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 $f(x)=\begin{cases} \frac{{2}^{x}+2}{2},x\leqslant 1 \\ |{\log }\_{2}(x-1)|,x > 1\end{cases}$, find the number of zeros of the function $F(x)=f[f(x)]-2f(x)-\frac{3}{2}$.

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