
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 the function $y=f\left(x\right)$ is an even function, when $x\geqslant 0$, $f\left(x\right)=x^{2}+2ax+1$ $(a$ is a constant).<br/>$(1)$ When $x \lt 0$, find the analytical expression of $f\left(x\right)$:<br/>$(2)$ Let the maximum value of the function $y=f\left(x\right)$ on $\left[0,5\right]$ be $g\left(a\right)$, find the expression of $g\left(a\right)$;<br/>$(3)$ For $g\left(a\right)$ in (2), find the set of all real numbers $m$ that satisfy $g\left(8m\right)=g(\frac{1}{m})$.

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