
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 following propositions:  
\\(①\\) The function \\(y=2\cos^2\left(\frac{1}{3}x+\frac{\pi}{4}\right)-1\\) is an odd function;  
\\(②\\) There exists a real number \\(\alpha\\) such that \\(\sin\alpha+\cos\alpha=\frac{3}{2}\\);  
\\(③\\) If \\(\alpha\\) and \\(\beta\\) are angles in the first quadrant and \\(\alpha < \beta\\), then \\(\tan\alpha < \tan\beta\\);  
\\(④\\) \\(x=\frac{\pi}{8}\\) is an equation of a symmetry axis for the function \\(y=\sin\left(2x+\frac{5\pi}{4}\right)\\);  
\\(⑤\\) The graph of the function \\(y=\sin\left(2x+\frac{\pi}{3}\right)\\) is symmetric about the point \\(\left(\frac{\pi}{12},0\right)\\).  
Among these propositions, the correct ones are \_\_\_\_\_\_ (fill in the numbers).

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