
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 $a$, $b$, $c$ represent three lines, and $\alpha$, $\beta$ represent two planes. Among the following propositions, identify the false statement(s):  
(1) If $c \perp \alpha$ and $c \perp \beta$, then $\alpha \parallel \beta$;  
(2) If $b \subset \alpha$, $c \nsubseteq \alpha$, and $c \parallel \alpha$, then $b \parallel c$;  
(3) If $b \subset \beta$, $c$ is the projection of $a$ onto $\beta$, and $b \perp c$, then $b \perp a$;  
(4) If $b \subset \beta$ and $b \perp \alpha$, then $\beta \perp \alpha$.

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