
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 the function $g\left(x\right)=mx^{2}-2mx+1+n$, where $\left(m \gt 0\right)$, has a maximum value of $0$ and a minimum value of $-1$ on the interval $\left[1,2\right]$.
$(1)$ Find the values of real numbers $m$ and $n$;
$(2)$ There exists $x\in \left[0,1\right]$, such that $g(2^{x})+1-k\cdot 2^{x+1}\geqslant 0$ holds. Find the range of real number $k$;
$(3)$ If $h\left(x\right)=\left(a-1\right)x^{2}+3x$, and $f\left(x\right)=g\left(x\right)+h\left(x\right)$, and for any $x\in \left[0,1\right]$, $|f\left(x\right)|\leqslant 1$, find the range of real number $a$.

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