Problem Description

When the RF chip of a 6G base station runs at an extremely high frequency band, it will generate a huge amount of heat. To ensure device safety, engineer Xiao Lan from the National Communications Laboratory designed an “intermittent transmission mechanism” to control heat dissipation.

The core logic of this mechanism is to switch the transmitter between the “working state” and the “sleep state”. The specific switching rule is determined by three key parameters set by Xiao Lan on the console: the start time $A$, the end time $M$, and an initial check value $S$.

The system clock $k$ will run from $2A$ to $M$. For each clock time $k$, the system decides whether it is currently “working” or “sleeping” according to the following logic:

1. Compute the period index $V$ corresponding to the current clock time $k$: $V = \lfloor \frac{k}{2} \rfloor - A + 1$ (note: $\lfloor x \rfloor$ means floor).

2. Determine the state:

   • If the period index $V$ is odd, then the chip is in the working state.
   • If the period index $V$ is even, then the chip enters the sleep state.

To verify whether the chip executes this switching logic exactly, the system will take all clock times $k$ that are in the working state, perform a bitwise XOR operation with the initial check value $S$, and call the resulting value the check code.

For example, let $A = 2, M = 7, S = 0$, then the range of $k$ is $4, 5, 6, 7$.

• $k = 4$: $V = \lfloor 4/2 \rfloor - 2 + 1 = 1$ (odd). State: working.
• $k = 5$: $V = \lfloor 5/2 \rfloor - 2 + 1 = 1$ (odd). State: working.
• $k = 6$: $V = \lfloor 6/2 \rfloor - 2 + 1 = 2$ (even). State: sleep (ignored).
• $k = 7$: $V = \lfloor 7/2 \rfloor - 2 + 1 = 2$ (even). State: sleep (ignored).

Check code $= S \oplus 4 \oplus 5 = 0 \oplus 4 \oplus 5 = 1$.

After confirming that the mechanism is correct, Xiao Lan entered the real extreme test parameters. Now, please help him compute the final check code:

• Start time $A = 12345678901235$
• End time $M = 98765432109874$
• Initial check value $S = 20260411$

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 the answer, only fill in this integer. Any extra content will result in no score.

Hint

Translated by ChatGPT 5