
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 $f\left(x\right)=a\sin 2x+b\cos 2x\left(ab\neq 0\right)$, there are four propositions below. Please identify the correct proposition(s) by filling in the blank with the corresponding number(s).

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

$\left(2\right)$ If $a=1, b=-1$, then the function $y=f\left(x\right)$ has a symmetry center at $\left( \frac{π}{4},0\right)$.

$\left(3\right)$ If $y=f\left(x\right)$ has a symmetry axis described by the equation $x= \frac{π}{8}$, then $a=b$.

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

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