
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 $y=\sin x+\cos x$, consider the following four statements:  
$(1)$ If $x\in[0,\pi]$, then $y\in[1, \sqrt{2}]$;  
$(2)$ The line $x= \frac{\pi}{4}$ is an axis of symmetry for the graph of the function $y=\sin x+\cos x$;  
$(3)$ On the interval $\left[\frac{\pi}{4}, \frac{5\pi}{4}\right]$, the function $y=\sin x+\cos x$ is increasing;  
$(4)$ The graph of the function $y=\sin x+\cos x$ can be obtained by shifting the graph of $y= \sqrt{2}\cos x$ to the right by $\frac{\pi}{4}$ units.  
Among these, the correct statements are \_\_\_\_\_\_.

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