
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 that $a$, $b$, $c$ are lines, and $\alpha$ is a plane, consider the following propositions:
1. If $a \parallel b$ and $b \perp c$, then $a \perp c$.
2. If $a \perp b$ and $b \perp c$, then $a \parallel c$.
3. If $a \parallel \alpha$ and $b \subset \alpha$, then $a \parallel b$.
4. If $a \perp \alpha$ and $b \subset \alpha$, then $a \perp b$.
5. If line $a$ and line $b$ are skew lines, then there can be at most one line perpendicular to both $a$ and $b$.
6. If $a \subset \alpha$, $b \subset \alpha$, $a \perp c$, and $b \perp c$, then $a \parallel b$.
The true propositions are __________.

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