
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
Let $f(x) = a\sin 2x + b\cos 2x$, where $a, b \in \mathbb{R}$, and $ab \neq 0$. If $f(x) \leq |f(\frac{\pi}{6})|$ holds for all $x \in \mathbb{R}$, then the correct conclusions are \_\_\_\_\_\_ (write down all the correct conclusion numbers).

① $f(\frac{5\pi}{12})=0$;  
② $|f(\frac{7\pi}{12})| \geq |f(\frac{\pi}{3})|$;  
③ The interval of monotonic increase for $f(x)$ is $(k\pi + \frac{\pi}{6}, k\pi + \frac{2\pi}{3})$ ($k \in \mathbb{Z}$);  
④ $f(x)$ is neither an odd function nor an even function.

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