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I would guess battery. Batteries can cause strange issues when they get weak. Plus, batteries have less cranking power when they are cold.
If it doesn't die once it is started it probably isn't your alternator.
I have seen several people on this site have problems with their batteries. Honda uses a very small group size battery and it doesn't have much cranking power. When it gets old it is much more likely to cause problems as it weakens because it was small to begin with.
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