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The tat tat tat sounds like the starter solenoid. Usually means that there's enough power to do things but not enough to start. It prevents damage to the cat by stopping the car turning over with no chance of starting.
Be very careful push starting a car - you really shouldn't push start cars fitted with cat converters as you end up with fuel being dumped into the cat which can overheat it very very quickly. An overheated cat + aluminimum / fibre glass car isn't good.
Once a battery completely discharges it'll start to form small crystals in the electolyte. This significantly reduces the capacity of the battery - It'll be able to hold volts but no amps (ie you'll be able to see 12-14v on a voltmeter but as soon as you try to start it it'll just not work).
The only options in this case is to get a automatic battery conditioner and leave it on for a day or two (you can get some that plug into the cigarette lighter socket as you drive and continue the job). These modulate the car's powersupply to breakdown the crystals and restore the battery back to health.
Other than that a new battery. Trying to charge a discharged battery doesn't usually work too well.
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