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

1. Find the analytic expression of this function;
2. Write the interval where this function is monotonically increasing;
3. Does there exist a real number $m$ that satisfies the inequality $A\sin (ω \sqrt {-m^{2}+2m+3}+φ) > A\sin (ω \sqrt {-m^{2}+4}+φ)$? If so, find the value(s) or range of $m$. If not, explain the reason.

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