
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 $f(x)=a\sin 2x+b\cos 2x (ab \neq 0)$, there are four propositions below. The sequence number of the correct proposition(s) is $\_\_\_\_\_\_$ (fill in the sequence number(s) of all correct propositions).

① If $a=1$, $b=-\sqrt{3}$, to obtain the graph of the function $y=f(x)$, simply shift the graph of the function $y=2\sin 2x$ to the right by $\frac{\pi}{6}$ units.

② If $a=1$, $b=-1$, then a symmetry center of the function $y=f(x)$ is $(\frac{\pi}{4}, 0)$.

③ If an equation of a symmetry axis of $y=f(x)$ is $x=\frac{\pi}{8}$, then $a=b$.

④ If the positive real roots of the equation $a\sin 2x + b\cos 2x = m$ form an arithmetic sequence in ascending order, then the common difference of this arithmetic sequence is $\pi$.

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