
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 $l$ be a line, and $\alpha$, $\beta$, $\gamma$ be distinct planes. Among the following statements, the false statement is __________.

(1) If $\alpha \perp \beta$, then there must exist a line within plane $\alpha$ that is parallel to plane $\beta$.

(2) If plane $\alpha$ is not perpendicular to plane $\beta$, then there can be no line within plane $\alpha$ that is perpendicular to plane $\beta$.

(3) If $\alpha \perp \gamma$ and $\beta \perp \gamma$, and $\alpha \cap \beta = l$, then $l \perp \gamma$.

(4) If $\alpha \perp \beta$, and $l$ intersects both planes $\alpha$ and $\beta$, then the angles formed between $l$ and planes $\alpha$ and $\beta$ are complementary.

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