
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
Let the function $f(x)=\log_{4}(4^{x}+1)+ax$ ($a\in\mathbb{R}$).

$(1)$ If $f(x)$ is an even function defined on $\mathbb{R}$, find the value of $a$;

$(2)$ If the inequality $f(x)+f(-x)\leqslant 2\log_{4}m$ holds for any $x\in[0,2]$, find the range of values for the positive real number $m$.

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