
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
Consider the sequences $(r_i),$ $(s_i),$ $(t_i)$ that represent, left to right, the elements in the 501st, 502nd, and 503rd rows of Pascal's triangle, respectively, with the leftmost element at $i = 0$. Compute the sum
\[
\sum_{i = 0; i \text{ even}}^{500} \frac{s_i}{t_i} - \sum_{i = 0; i \text{ even}}^{500} \frac{r_i}{s_i}.
\]

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