
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 α and β are planes, and m and n are lines, consider the following propositions:

1. If line m is perpendicular to plane α and m is contained in plane β, then plane α is perpendicular to plane β.
2. If line m is contained in plane α and line n is also contained in plane α, and if m is parallel to plane β and n is parallel to plane β, then plane α is parallel to plane β.
3. If line m is contained in plane α, line n is not contained in plane α, and m and n are skew lines, then line n intersects plane α.
4. If the intersection of planes α and β is line m, and line n is parallel to m, and n is not contained in α nor β, then line n is parallel to both α and β.

The correct propositions are (fill in the correct proposition numbers).

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