
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 $f(x)=a\sin 2x+b\cos 2x\ (ab\ne 0)$, there are four propositions below. Fill in the correct proposition numbers.

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

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

3. If an axis of symmetry of $y=f(x)$ is $x=\dfrac{\pi}{8}$, 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}
user