
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 $g(x)=\frac{4^x-n}{2^x}$ is an odd function, and $f(x)=\log_4(4^x+1)+mx$ is an even function.

$(1)$ Find the value of $m+n$;

$(2)$ Let $h(x)=f(x)+\frac{1}{2}x$, if $g(x) > h\left[ \log_4(2a+1) \right]$ holds for any $x \in [1,+\infty)$, find the range of the real number $a$.

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