
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
The quadratic function $g\left(x\right)=mx^{2}-2mx+n+1\left(m \gt 0\right)$ has a maximum value of $4$ and a minimum value of $0$ on the interval $\left[0,3\right]$. <br/>$(1)$ Find the analytical expression of the function $g\left(x\right)$; <br/>$(2)$ Let $f\left(x\right)=g\left(x\right)+\left(2-a\right)x$, and the minimum value of $f\left(x\right)$ on $\left[-1,2\right]$ is $-3$, find the value of $a$.

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