
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. There exists $$\alpha \in (0, \frac{\pi}{2})$$ such that $$\sin \alpha + \cos \alpha = \frac{1}{3}$$;
2. There exists an interval $(a, b)$ such that $y = \cos x$ is a decreasing function and $\sin x < 0$;
3. $y = \tan x$ is an increasing function within its domain;
4. $$y = \cos 2x + \sin\left(\frac{\pi}{2} - x\right)$$ has both maximum and minimum values and is an even function;
5. $$y = \sin\left|2x + \frac{\pi}{6}\right|$$ has the smallest positive period of $\pi$.

The correct statements are $\boxed{\text{answer}}$.

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