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

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