
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
Given the line system $M$: $x\cos θ+(y-1)\sin θ=1(0≤θ≤2π)$, determine the correct statement(s) among the following:

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

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