
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 $f(x)=m{({\frac{1}{2}})^{|x|}}+n$ passes through the origin and approaches the line $y=1$ infinitely close but does not intersect it.  
$(1)$ Find the analytical expression of $f\left(x\right)$.  
$(2)$ For the function $g(x)={({\frac{1}{4}})^x}+f(x)$, $x\in \left[0,2\right]$, find the minimum value of $g\left(x\right)$.

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