Problem Description

Xiao Lan bought a smart toy frog. Today, he put this toy frog on a straight track and started the program to make it jump forward.

Because the battery power is being used up, the distance of each jump decreases regularly. It is known that the distance of its $i$-th jump $d_i$ (unit: meters) is exactly:

$$d_i = \frac{1}{(2i-1)(2i+1)}$$

After it completed $2026$ consecutive jumps, the toy frog stopped because the battery was exhausted. The total distance it moved forward is $S$ (unit: meters):

$$S = \frac{1}{1 \times 3} + \frac{1}{3 \times 5} + \frac{1}{5 \times 7} + \dots + \frac{1}{4051 \times 4053}$$

Now express $S$ as an irreducible fraction $\frac{x}{y}$ (where $x$ and $y$ are coprime). Please help Xiao Lan find the value of $x+y$ (if the computed total distance is $\frac{12}{18}$, after reducing it to an irreducible fraction it becomes $\frac{2}{3}$, then $x=2, y=3$, and the output should be $2+3=5$).

Output Format

This is an output-only fill-in-the-blank problem. You only need to compute the result and submit it. The result of this problem is an integer. When submitting, you only need to write a program that outputs this integer. Any extra content will make you score no points.

Hint

Translated by ChatGPT 5