
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
The function $y=A\sin (\omega x+\varphi)$ ($A > 0$, $\omega > 0$, $0 \leqslant \varphi \leqslant \frac {\pi}{2}$) takes only one maximum value and one minimum value in the interval $x\in(0,7\pi)$, and when $x=\pi$, $y_{max}=3$; when $x=6\pi$, $y_{min}=-3$.
1. Find the analytical expression of this function;
2. Find the interval of monotonic increase for this function;
3. Determine whether there exists a real number $m$ that satisfies the inequality $A\sin (\omega \sqrt {-m^{2}+2m+3}+\varphi) > A\sin (\omega \sqrt {-m^{2}+4}+\varphi)$. If it exists, find the range (or value) of $m$; if not, explain why.

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