Let's break the problem into two phases: before and after $Y$ changes direction.

Phase 1: Running in the same direction

• We are given that when $X$ completes $7$ rounds, $Y$ completes exactly $5$ rounds.
• Let this duration be $T$.
• The speeds of $X$ and $Y$ are $V_X = 7$ rounds per $T$ and $V_Y = 5$ rounds per $T$.
• Since they are running in the same direction, their relative speed is $V_X - V_Y = 7 - 5 = 2$ rounds per $T$.
• The number of times they meet is equal to the relative distance covered in rounds. In time $T$, they cover a relative distance of $2$ rounds, meaning they meet exactly $2$ times.
• These meetings occur at $t = 0.5T$ and $t = 1.0T$. The meeting at $t = 1.0T$ happens exactly when $X$ completes $7$ rounds and $Y$ completes $5$ rounds (both are at the starting point).

Phase 2: Running in opposite directions

• After $t = 1.0T$, $Y$ changes direction and doubles his speed.
• $Y$'s new speed is $V_Y' = 2 \times 5 = 10$ rounds per $T$.
• $X$ continues in the same direction with the same speed, $V_X = 7$ rounds per $T$.
• Because they are now running in opposite directions, their new relative speed is $V_X + V_Y' = 7 + 10 = 17$ rounds per $T$.
• $X$ stops when he completes exactly $21$ rounds. Since he already completed $7$ rounds in Phase 1, he must run $21 - 7 = 14$ more rounds in Phase 2.
• The time taken for Phase 2 is the distance $X$ runs divided by his speed: $\frac{14 \text{ rounds}}{7 \text{ rounds per } T} = 2T$.
• In this duration of $2T$, the number of times they meet is their relative speed multiplied by the time: $17 \times 2 = 34$ times.
• These $34$ meetings happen strictly after $t = 1.0T$. The very last meeting (the 34th in this phase) occurs exactly at the end of the $2T$ duration (at $t = 3.0T$), which is when $X$ completes his 21st round and they both finally stop.

Total Number of Meetings:

• The question asks for the number of meetings after they had started and before they finally stopped.
• We exclude the start ($t = 0$) and the final stop ($t = 3.0T$).
• Meetings in Phase 1: $2$ (at $t = 0.5T$ and $t = 1.0T$).
• Meetings in Phase 2: $34 - 1 = 33$ (excluding the final stop at $t = 3.0T$).
• Total valid meetings = $2 + 33 = 35$.

Therefore, $X$ and $Y$ met $35$ times.