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The SYS indication is produced by the CCM and is module 1 in dianoistic test mode. Many times it means the battery needs to be replaced. But before we draw conclusions, do you have any indication that the battery is week or bad. Check the digital voltmeter indication (with the gauge button) what does the volts indicate? (watch for a while)
It would also be a good idea to go into the dianostic mode and check for any error codes and what module it is from. Mod 1 is CCM and Mod 4 is ECM. Not sure if your flashing light is from a code or just because the engine died.
It is unlikly (but possible) that the battery would kill a running engine and would need to have as shorted cell or have a bad problem with it. But the battery can act up to cause the SYS to flash.
I'm lucky to get 3 years out of a battery in FL because of the heat. The vette (electronics) needs a good battery and is sensitive to the condition of it.