
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 two distinct lines $a,b$ and two distinct planes $\alpha, \beta$, determine the truth value of the following propositions:

(1) If $a \bot \alpha$ and $a \subset \beta$, then $\alpha \bot \beta$;

(2) If $a \parallel \alpha$ and $\alpha \bot \beta$, then $a \bot \beta$;

(3) If $a \bot \beta$ and $\alpha \bot \beta$, then $a \parallel \alpha$;

(4) If $a \bot \alpha$ and $b \bot \alpha$, then $a \parallel b$.

The serial number(s) of all true propositions is/are \_\_\_\_\_\_\_.

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