
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 line $l$, $m$ and plane $\alpha$, $\beta$, determine the correct propositions among the following:
  1. If $l$ is parallel to a line in $\alpha$, then $l \parallel \alpha$;
  2. If $l$ is perpendicular to two lines in $\alpha$, then $l \perp \alpha$;
  3. If $l \parallel \alpha$, $l \subset \beta$ and $\alpha \cap \beta = m$, then $l \parallel m$;
  4. If $m \subset \alpha$, $l \subset \beta$ and $l \perp m$, then $\alpha \perp \beta$;
  5. If $m \subset \alpha$, $l \subset \alpha$, and $m \parallel \beta$, $l \parallel \beta$, then $\alpha \parallel \beta$;
  6. If $\alpha \parallel \beta$, $\alpha \cap \gamma = l$, $\beta \cap \gamma = m$, then $l \parallel m$;

Provide the numbers of all correct propositions.

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