
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 that the vertex of the quadratic function $f(x)$ is $A(1,16)$, and the graph intersects the $x$-axis forming a segment of length $8$.

$(1)$ Find the analytical expression of the function $f(x)$.

$(2)$ Let $g(x)=(2-2a)x-f(x)$, if the function $g(x)$ is monotonically increasing on $x \in [0,2]$, find the range of the real number $a$.

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