
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 a quadratic function $f(x)$ has two intersections with the x-axis at $(-3,0)$ and $(5,0)$, and $f(2)=15$.

$(1)$ Find the expression for the function $f(x)$;

$(2)$ Let $g(x)=(2-2m)x-f(x)$, find the minimum value of the function $g(x)$ when $x \in [0,2]$.

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