
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 $g(x)=ax^{2}-2ax+1+b\ (a > 0)$ defined on the interval $[2,4]$. Its maximum value is $9$ and the minimum value is $1$. Let $f(x)=g(|x|)$.
(1) Find the values of real numbers $a$ and $b$.
(2) If the inequality $f(\log_{2}k) > f(2)$ holds true, find the range of real number $k$.

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