
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
Let $\alpha$ represent a plane, and $a$, $b$ represent lines. Consider the following four propositions:
1. If $a \parallel b$ and $a \perp \alpha$, then $b \perp \alpha$.
2. If $a \perp \alpha$ and $b \perp \alpha$, then $a \parallel b$.
3. If $a \perp \alpha$ and $a \perp b$, then $b \parallel \alpha$.
4. If $a \parallel \alpha$ and $a \perp b$, then $b \perp \alpha$.

Identify the correct propositions. (Fill in the numbers of all correct propositions)

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