
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 $f\left(x\right)=ax^{2}+bx+c$, where $f\left(0\right)=1$, $f\left(1\right)=0$, and $f\left(x\right)\geqslant 0$ for all real numbers $x$. <br/>$(1)$ Find the analytical expression of the function $f\left(x\right)$; <br/>$(2)$ If the maximum value of the function $g\left(x\right)=f\left(x\right)+2\left(1-m\right)x$ over $x\in \left[-2,5\right]$ is $13$, determine the value of the real number $m$.

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