
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 defined on \(\mathbf{R} \), with \( f(1) = 1 \), and for any \( x < 0 \), it holds that
$$
f\left( \frac{x}{x-1} \right) = x f(x).
$$

Find the value of \( \sum_{i=1}^{50} f\left( \frac{1}{i} \right) f\left( \frac{1}{101-i} \right) \).

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