
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 $f(x)\leqslant |f\left( \frac{\pi }{6} \right)|$ holds for all $x\in \mathbb{R}$, then which of the following statements are correct:

1. $f\left( \frac{11\pi }{12} \right)=0$;
2. $|f\left( \frac{7\pi }{12} \right)| < |f\left( \frac{\pi }{5} \right)|$;
3. $f(x)$ is neither an odd function nor an even function;
4. The interval of monotonic increase for $f(x)$ is $\left[ k\pi +\frac{\pi }{6},\,k\pi +\frac{2\pi }{3} \right]$, $(k\in \mathbb{Z})$;
5. All lines passing through the point $(a,b)$ intersect with the graph of the function $f(x)$.

Write down the numbers of all correct statements.

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