Actual chances aside and hypothetically: how does society respond in the first 1-3 months of confirmed aliens?

Feels like there would be a large existential crisis to some extent.

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Absolutely. 

I don't expect first encounter would result in immediate destruction of life on earth. 

There will be tons of chaos and fear in the immediate aftermath. being in a position to help people process and not devolve further into violence will require material capital and reach.