
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 $a$ and $b$ be positive numbers. Determine the validity of the following propositions:
1. If $a^{2}-b^{2}=1$, then $a-b < 1$;
2. If $\frac{1}{b}-\frac{1}{a}=1$, then $a-b < 1$;
3. If $e^{a}-e^{b}=1$, then $a-b < 1$;
4. If $\ln{a}-\ln{b}=1$, then $a-b < 1$.

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