
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 quadratic function $f(x)=ax^{2}-4x+2$, and the function $g(x)=(\frac{1}{3})^{f(x)}$.

(1) If $f(2-x)=f(2+x)$, find the analytic expression of $f(x)$;

(2) If $g(x)$ has a maximum value of $9$, find the value of $a$ and the range of $g(x)$;

(3) Given that $a\leqslant 1$, if the function $y=f(x)-\log_{2}\frac{x}{8}$ has only one zero point in the interval $[1,2]$, determine the range of the real number $a$.

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