My iPhone 4s Won't Turn On Past Apple Logo?

Okay, so the other day, my phone was low and it died. Later on, I put it on charge and after a while it began booting up, as normal. It displayed the Apple Logo for like 2 minutes, and then just cut out. I thought that was weird, but just assumed that maybe I'd knocked the power cable when I picked it up. I took out the cable and put it back in, and it began turning on immediately. Again, for about 2 minutes, it displayed the Apple Logo, then cut out. I tried it multiple times, but it just did the same again and again. I did all the different ways of turning it on and stuff, nothing worked. I tried using a different charger, still the same result. I took it to a shop and they replaced the battery with one I'd bought earlier that year, still wouldn't turn on. They replaced the battery with one of theirs. STILL wouldn't turn on. They tried 'flashing' it, but when I came back the next day, again, nothing. I'm getting really frustrated because my phone is simply a part of my daily routine, and it's important. I don't have the money to buy a new one, and won't for literally at least a year. (I pay for all my things with pocket money, £12 a week). My phone isn't on contract, I bought it refurbished and it's on Pay As You Go. I don't know what else I can do. Please help! Any suggestions at all will be appreciated, Thanks.
One more thing, my phone up to this point has been in immaculate condition. It's never been dropped, had water damage, or endured any other form of mistreatment.

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