
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
We call a number antitriangular if it can be expressed in the form \(\frac{2}{n(n+1)}\) for some natural number \(n\). For how many numbers \(k\) (where \(1000 \leq k \leq 2000\)) can the number 1 be expressed as the sum of \(k\) antitriangular numbers (not necessarily distinct)?

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