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Today, I spent time and installed Windows XP SP2 from scratch. The Heroes 3: The Restoration of Erathia game ran normally without any problems. I gradually installed drivers for the motherboard, video card, and sound card, installed the firewall and antivirus that I had before, and updated DirectX to rule them out. The game continued to run normally. It seems something was wrong with the system, and something was probably conflicting.
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Reinstalling Windows helped, but only a little; the game still doesn't want to launch properly, meaning it can be launched successfully only once every two or three tries. THE QUESTION REMAINS OPEN. :(