cinders, ashes, embers, remnants
烬 centers on the remains left after fire has consumed something: cinders, ashes, embers, or the remnants of what was burned.
烬 combines 火 (fire) with 尽, which likely contributes the sound and the idea of something being exhausted or used up, together suggesting the remains after burning.
Fire 火 on the left and 尽 (exhaust) on the right: imagine a fire burning until nothing is left but cinders and ashes.
For ジン, picture a gin bottle tossed into the fire, leaving only cinders behind: gin -> ジン.
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