
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 $f(x)=\begin{cases} \frac{{{2}^{x}}+2}{2},x\leqslant 1 \\ \left| {{\log }\_{2}}(x-1) \right|,x > 1 \end{cases}$, 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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