
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 $f(x)=\sin \omega x\cos \omega x-\sqrt{3}\sin^{2}\omega x$ with $\omega > 0$ and a minimum positive period of $\pi$, consider the following statements:<br/>① $\omega =1$;<br/>② The monotonically increasing interval of $f(x)$ is $[-\frac{5\pi}{12}+2k\pi,\frac{\pi}{12}+2k\pi]$, where $k\in \mathbb{Z}$;<br/>③ The graph of $f(x)$ is symmetric about the $y$-axis after shifting it to the left by $\frac{\pi}{12}$ units;<br/>④ $f(\frac{\pi}{3}+x)+f(\frac{\pi}{3}-x)=-\sqrt{3}$.<br/>The correct statements are ____.

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