
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
Among the following four propositions:  
(1) If $\alpha > \beta$, then $\sin\alpha > \sin\beta$  
(2) The negation of the proposition "For all $x > 1$, $x^2 > 1$" is "There exists $x \leq 1$, $x^2 \leq 1$"  
(3) The necessary and sufficient condition for the lines $ax + y + 2 = 0$ and $ax - y + 4 = 0$ to be perpendicular is $a = \pm1$  
(4) The contrapositive of the proposition "If $xy = 0$, then $x = 0$ or $y = 0$" is "If $x \neq 0$ or $y \neq 0$, then $xy \neq 0$"  
The correct proposition number is a key point: negation of a proposition, contrapositive, necessary and sufficient condition.

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