
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 $\alpha \in (0, \pi)$ such that $\sin \alpha + \cos \alpha = 1$;  
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. $y = \cos 2x + \sin (-x)$ has both maximum and minimum values and is an even function;  
5. The smallest positive period of $y = |\sin (2x + \pi)|$ is $\pi$,  
Among the above statements, the incorrect ones are $\boxed{1, 2, 3, 5}$.

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