
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 $f\left(x\right)=\log _{2}(2^{x}+1)+ax$ is an even function.<br/>$(1)$ Find the value of $a$;<br/>$(2)$ Let $g\left(x\right)=f\left(x\right)+x$, $h\left(x\right)=x^{2}-2x+m$. If for any $x_{1}\in \left[0,4\right]$, there exists $x_{2}\in \left[0,5\right]$ such that $g(x_{1})\geqslant h(x_{2})$, find the range of values for $m$.

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