
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 functions $f(x)=\sin x+\cos x$ and $g(x)=\sin 2x-f(x)$.  
$(1)$ Find the minimum positive period and axis of symmetry equation of the function $f(x)=\sin x+\cos x$;  
$(2)$ When $x∈[{-\frac{π}{2},0}]$, find the range of the function $g(x)$;  
$(3)$ Let $h(x)=\frac{{9^x}-1}{{9^x}+1}$. When $x\in \left(0,+\infty \right)$, the inequality $mh({\frac{x}{2}})-h(x)＞0$ always holds. The set corresponding to the range of real number $m$ is denoted as $M$. If under the conditions of $(1)$, $ag\left(x\right)\notin M$ always holds (where $a \gt 0$), find the range of real number $a$.

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