
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
Among the following statements:
① The perimeter of a sector is 8cm, and its area is 4cm<sup>2</sup>. Then, the central angle of the sector in radians is 1rad;
② The maximum value of the function $f(x) = 2\cos(x)(\sin(x) + \cos(x))$ is $\sqrt{2}$;
③ If $\alpha$ is an angle in the third quadrant, then the value of $$y= \frac{|\sin \frac{\alpha}{2}|}{\sin \frac{\alpha}{2}} + \frac{|\cos \frac{\alpha}{2}|}{\cos \frac{\alpha}{2}}$$ is either 0 or -2;
④ If $\sin(\alpha) = \sin(\beta)$, then the terminal sides of $\alpha$ and $\beta$ are the same;
⑤ The function $$f(x)= \begin{cases} 0, & \text{if } x \text{ is rational} \\ 1, & \text{if } x \text{ is irrational} \end{cases}$$ is a periodic function;
The correct statements are __________ (Write down all correct answers).

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