
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 $f\left(x\right)=a^{x}$ where $a \gt 0$ and $a\neq 1$ passes through the point $\left(4,4\right)$.  
$(1)$ Find the value of $a$;  
$(2)$ Compare $f\left(-2\right)$ with $f(m^{2}-2m)$ where $m\in \mathbb{R}$;  
$(3)$ Find the range of the function $g\left(x\right)=a^{|x-1|}$ for $-3\leqslant x\leqslant 3$.

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