
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 odd function defined on $R$, $g\left(x\right)$ is an even function defined on $R$, and $f\left(x\right)+g\left(x\right)=e^{-x}$.  
$(1)$ Find the analytical expressions for functions $f\left(x\right)$ and $g\left(x\right)$.  
$(2)$ The inequality with respect to $x$, $2g\left(x\right)+a(e^{-x}-1)\geqslant 1$, holds for all $x$ in $\left(0,+\infty \right)$, find the range of real number $a$.

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