
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 $f\left(x\right)=a^{x}$ ($a \gt 0$, and $a\neq 1$).  
$(1)$ Determine the parity and monotonicity of the function $h\left(x\right)=f\left(x\right)-f\left(-x\right)$, and provide a proof.  
$(2)$ Find the range of the function $g\left(x\right)=f\left(x\right)+f\left(-x\right)$.

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