
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 $a > b$, then $\dfrac{a}{b} > \dfrac{b+1}{a-1}$;

2. If $a > b > c > 0$, then $\dfrac{a-c}{c} > \dfrac{b-c}{b}$;

3. If $a < b < 0$, then $\dfrac{1}{a-b} > \dfrac{1}{a}$;

4. If $a > |b|$, then $a^{2} > b^{2}$

Which of the above propositions are true?

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