
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 $a, b \in \mathbb{R}, ab \neq 0$. If $f(x) \leqslant |f(\frac{\pi}{6})|$ holds for all $x \in \mathbb{R}$, then the correct conclusions are:

1. $f(\frac{11\pi}{12}) = 0$.
2. $f(x)$ is neither an odd function nor an even function.
3. $|f(\frac{7\pi}{10})| < |f(\frac{\pi}{5})|$.
4. There exists a line passing through the point $(a, b)$ that does not intersect the graph of $f(x)$.
5. When $b > 0$, the interval of monotonic increase for $f(x)$ is $\left[-\frac{\pi}{3} + k\pi, \frac{\pi}{6} + k\pi\right] (k \in \mathbb{Z})$.

Write the numbers of the correct conclusions.

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