
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 that $f\left(x\right)$ is an even function and $g\left(x\right)$ is an odd function, and they satisfy $f\left(x\right)-g\left(x\right)=2^{1-x}$.  
$(1)$ Find $f\left(x\right)$ and $g\left(x)$;  
$(2)$ If the equation $mf\left(x\right)=\left[g\left(x\right)\right]^{2}+2m+9$ has a solution, find the range of real numbers for $m$.

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