
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 \( f(x) \) is an odd function on \(\mathbf{R}\), \( f(1) = 1 \), and for any \( x < 0 \), \( f\left(\frac{x}{x-1}\right) = x f(x) \). Find the value of \( f(1) f\left(\frac{1}{100}\right) + f\left(\frac{1}{2}\right) f\left(\frac{1}{99}\right) + f\left(\frac{1}{3}\right) f\left(\frac{1}{98}\right) + \cdots + f\left(\frac{1}{50}\right) f\left(\frac{1}{51}\right) \).

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