
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 that $a > 0 > b > -a$ and $c < d < 0$, determine the correct propositions among the following:
$(1) \; ad > bc$; $(2) \; \frac{a}{b} + \frac{b}{c} < 0$; $(3) \; a - c > b - d$; $(4) \; a(d - c) > b(d - c)$.

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