
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
(1) The domain of the function $y=\ln(2\sin x-\sqrt{2})$ is _______.

(2) Given that the function $f(x)=x^{2}-2x\sin\theta+1$ has a root, the set of values for $\theta$ is _______.

(3) If line $l_{1}: 6x+my-1=0$ is parallel to line $l_{2}: 2x-y+1=0$, then $m=$ _______.

(4) For the function $f(x)=2\sin(x+\frac{\pi}{4})$, consider the following four propositions:  
① There exists $\alpha\in(-\frac{\pi}{2},0)$ such that $f(\alpha)=\sqrt{2}$;  
② There exists $\alpha\in(0,\frac{\pi}{2})$ such that $f(x-\alpha)=f(x+\alpha)$ always holds;  
③ There exists $\alpha\in\mathbb{R}$ such that the graph of $f(x+\alpha)$ is centrally symmetric about the origin;  
④ The graph of $f(x)$ is symmetric about the line $x=-\frac{3\pi}{4}$;  
⑤ The graph of $f(x)$ can be obtained by shifting the graph of $y=-2\cos x$ to the left by $\frac{\pi}{4}$ units.  
Among these, the correct proposition numbers are _______.

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