This isn't a job interview - they're not going to ask you why you have gaps in your education. And if they did, two highly prestigious internships would certainly be reason enough.
And yes, you're overthinking it. Relax, look into informal rush opportunities as soon as you start at school - they won't necessarily be advertised but the Greek Life office should be able to help get you started, and stop worrying about it so much until then.
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"Traveling - It leaves you speechless, then turns you into a storyteller. ~ Ibn Battuta
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