
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
A function $y=A\sin (ωx+φ)(A > 0,ω > 0)$ reaches a maximum and a minimum value within the interval $x∈(0,7π)$. When $x=π$, the maximum value of $y$ is $3$, and when $x=6π$, the minimum value of $y$ is $-3$.

(1) Find the analytical expression of this function;

(2) Determine if there exists a real number $ω$ that satisfies $A\sin (ω\sqrt{-{m}^{2}+2m+3} +φ) > A\sin (ω\sqrt{-{m}^{2}+4} +φ)$. If it exists, find $m$. If not, explain the reason.

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