
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 \( b\left[\frac{1}{1 \times 3}+\frac{1}{3 \times 5}+\cdots+\frac{1}{1999 \times 2001}\right]=2 \times\left[\frac{1^{2}}{1 \times 3}+\frac{2^{2}}{3 \times 5}+\cdots+\frac{1000^{2}}{1999 \times 2001}\right] \), find the value of \( b \).

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