
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
The graph of the function $y=\sin 2x$ is translated to the left by $\frac{\pi}{6}$ units along the $x$-axis and the vertical coordinates are stretched to twice their original length (the horizontal coordinates remain unchanged), resulting in the graph of the function $y=f(x)$. The following four statements are given regarding the function $y=f(x)$:

1. The analytical expression of the function is $y=2\sin (2x+\frac{\pi}{6})$;
2. The graph of the function is symmetric about the point $(\frac{\pi}{3},0)$;
3. The function is increasing on the interval $[0,\frac{\pi}{6}]$;
4. The minimum value of the function $y=f(x)+a$ on the interval $[0,\frac{\pi}{2}]$ is $\sqrt{3}$, therefore $a=2\sqrt{3}$.

Determine the sequence number(s) of the correct statement(s).

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