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

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