How to Run synchronous AC motor?

Hi all. I do not know about motors at all. I just know how to run stepper motor.

I have a laminator which i am going to modify it for PCB. But i think i have figured out (not sure)how the motor is working so need your confirmation.

This is the motor i am dealing with

Kysan Electronics Product Detail Page

Everything is same except voltage rated is 220V.

It has 4 wires --RED--BLUE--BLUE--YELLOW

Both blue wires are soldered together and fed to 220V. There is a capacitor (capacitor value = 630V274J ) whose one leg is connected to RED and one connected to YELLOW. Both RED and YELLOW wires are fed in to 3 terminal switch 1---2---3
2 is connected to 220V and 1 is connected to RED and 3 is connected to YELLOW and this three terminal switch makes the motor forward and reverse.

So i think i have guessed it right, just i have to connect 2 blue wires to 220 and give 220V either to RED or Yellow to run it in specific direction with the capacitor installed, right?

And what is the role that capacitor is playing?

You have already answered "how to run", so the remaining question is only "how does it work". This synchronous motor animation shows a two-phase capacitor motor:
Electric motors and generators