
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 $f(x)=A\sin (ωx+φ)$, where $A > 0,ω > 0,0 < φ < π$, and the smallest positive period of the function $f(x)$ is $\frac{π}{2}$.

(1) If the function $f(x)$ takes its minimum value $(-2)$ at $x=\frac{π}{3}$, find the analytic expression of the function $f(x)$;

(2) If all points on the function graph $f(x)$ are horizontally stretched to twice their original length (the vertical coordinates remain unchanged), and then shifted to the left by $\frac{π}{6}$ units, the resulting function graph is symmetric about the $y$-axis. Find the monotonically increasing interval of the function $f(x)$.

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