IR Demodulator

It takes a 32. 768 kHz IR input and outputs a logic low when the input is present. I see 3 distinct stages. everything up to U1A, everything between U1A and Q3 and everything to the right of Q3. I follow the left and the right. but not the center which is where all the magic happens. Can anyone explain the piece in the middle It makes no sense to me. How is this circuit functioning Why do they make use of all the gates
IR Demodulator - schematic

Why do they have a resistor in series with Vcc on U1 What purpose does R26 serve

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