
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 a function $f(x)$ that satisfies $\forall x\in R$, $f(x)+2f(-x)=2^{x}+2^{-x}$.<br/>$(1)$ Find the analytical expression of $f(x)$;<br/>$(2)$ If $a \gt 0$, and the function $g(x)=\frac{x+5}{4}$, and for all $m\in \left[1,2\right]$, there exists $n\in [2\sqrt{2}-5,\frac{2}{3}]$ such that $f(am)=g(n)$, find the range of real number $a$.

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