
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 $f\left(x\right)=ax^{2}+bx+c\left(a\neq 0\right)$, and it satisfies the following conditions: ① the solution set of the inequality $f\left(x\right) \lt 0$ is $\{x\left|\right.1 \lt x \lt 2\}$; ② the graph of the function $y=f\left(x\right)$ passes through the point $\left(3,2\right)$.
$(1)$ Find the analytical expression of the function $f\left(x\right)$;
$(2)$ Let $g\left(x\right)=f\left(x\right)-mx$, find the minimum value $h\left(m\right)$ of the function $g\left(x\right)$ on the interval $\left[1,2\right]$.

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