
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 functions $f(x)=5+\log_{\frac{1}{2}}(-x^{2}+2x+7)$, $x\in[0,3]$, and $g(x)=9^{x}-2m\cdot 3^{x}$, $x\in[0,1]$.
$(1)$ Find the monotonic intervals and extreme values of $f(x)$.
$(2)$ Let the range of $f(x)$ be $M$ and the range of $g(x)$ be $N$. Is there a real number $m$ such that $M\subseteq N$? If so, find the range of the real number $m$; if not, explain the reason.

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