
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
Consider the family of lines M: $x\cos\theta + (y-1)\sin\theta = 1$ ($0 \leq \theta \leq 2\pi$). Regarding the following statements:

(1) All lines in M pass through a fixed point;
(2) There exists a circle that does not intersect with any line in M;
(3) For any integer $n$ ($n \geq 3$), there exists a regular $n$-sided polygon, all of whose sides lie on lines in M;
(4) The area of the equilateral triangles that can be formed by lines in M are all equal.

Among these statements, the correct ones are \_\_\_\_\_\_ (fill in the sequence number).

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