
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 following four propositions concerning lines $a$, $b$ and planes $\alpha$, $\beta$:
(1) If $a \perp b$ and $b \perp \alpha$, then $a \parallel \alpha$;
(2) If $a \parallel \alpha$, $\alpha \parallel \beta$, and $b \perp \beta$, then $a \perp b$;
(3) If $a \not\subset \alpha$, $a \parallel b$, and $b \subset \alpha$, then $a \parallel \alpha$;
(4) If $a \parallel \alpha$ and $\alpha \perp \beta$, then $a \perp \beta$.
Identify the sequence number(s) of all correct propositions.

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