
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 $f(x)=\frac{{4}^{x}}{{4}^{x}+2}+a$, and $f\left(\lg 2\right)+f\left(\lg 5\right)=3$.  
$(1)$ Find the value of $a$;  
$(2)$ For $x\in \left[-1,1\right]$, if $f\left(x\right)\geqslant 4^{x}+m$ always holds, find the range of $m$.

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