
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
Regarding the function $f(x)=\sin 2x-\cos 2x$, the following propositions are given:

$(1)$ The period of the function $y=f(x)$ is $\pi$;

$(2)$ The line $x=\frac{\pi}{4}$ is an axis of symmetry for $y=f(x)$;

$(3)$ The point $\left(\frac{\pi}{8},0\right)$ is a center of symmetry for the graph of $y=f(x)$;

$(4)$ The graph of $y=f(x)$ can be obtained by shifting the graph of $y=\sqrt{2}\sin 2x$ to the left by $\frac{\pi}{4}$ units,

Among these, the correct propositions are ______________ (Write down the numbers of all propositions you believe to be true).

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