
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 $y=A\sin (\omega x+\varphi)$ ($A > 0$, $\omega > 0$, $0 \leqslant \varphi \leqslant \frac {\pi}{2}$) can only reach one maximum value and one minimum value within $x \in (0,9\pi)$, and when $x=\pi$, $y$ reaches its maximum value of $4$, and when $x=8\pi$, $y$ reaches its minimum value of $-4$.
\\((1)\\) Find the expression of this function and its interval of monotonic increase;
\\((2)\\) Is there a real number $m$ that satisfies the inequality $A\sin (\omega \sqrt {m+1}+\varphi) > A\sin (\omega \sqrt {-m+4}+\varphi)$? If it exists, find the range of $m$; if not, explain why.

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