
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
For the function $f(x) = 2\cos^2x + 2\sin x\cos x - 1$ ($x \in \mathbb{R}$), the following statements are given:
1. The smallest positive period of $f(x)$ is $2\pi$.
2. $f(x)$ is a decreasing function in the interval $[\,]$.
3. The line $x =$ is a line of symmetry for the graph of $f(x)$.
4. The graph of $f(x)$ can be obtained by shifting the graph of $y = \sin 2x$ to the left.

Among these statements, the correct ones are \_\_\_\_\_\_\_\_ (fill in all that you think are correct).

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