
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 \( f(x) \) is an odd function defined on \( \mathbf{R} \), \( f(1) = 1 \), and for any \( x < 0 \), it holds that
\[ 
f\left(\frac{1}{x}\right) = x f\left(\frac{1}{1-x}\right) . 
\]
Then find the value of the sum
\[ 
\sum_{k=1}^{1009} f\left(\frac{1}{k}\right) f\left(\frac{1}{2019-k}\right).
\]

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