
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
An irreducible fraction \(\frac{a}{b}\) is defined by the equation

$$
\frac{a}{b}=\frac{999}{1999} + \frac{999}{1999} \cdot \frac{998}{1998} + \frac{999}{1999} \cdot \frac{998}{1998} \cdot \frac{997}{1997} + \ldots + \frac{999}{1999} \cdot \frac{998}{1998} \cdot \ldots \cdot \frac{1}{1001}
$$

Find \(a\) and \(b\).

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