Can't get my google nexus 4 to start. Please help

kshitizarora80895

New to this forum
Hello, I am a nexus 4 user and has been using it for 2 years now. But suddenly my nexus 4 isn't able to turn on. I had it working well yesterday afternoon it had an update which was not able to install as there was less than 500mb of space left. I kept the phone in my pocket and left home, after some time i realised it was switched off i tried turning it on but sometimes it cannot get past the google logo and sometimes it gets past and tries to start but a screen of optimizing apps shows up every time it manages to reach so far then it optimise 240 apps, I would also like to mention one more thing the phone is getting switched off when moved or shaken maybe it has something to do with battery.
I would also like to inform you some previously similar issues i faced earlier, i have used my nexus very rough so there has been a screen replacement and a battery replacement, it has many roughed and broken edges but it works fine when it does. The problem i faced before was that,my phone was on charge and when i took it out it used to get switched off. I realised the battery inside might be loose. but it happened for some days then it started working fine again.
I am confused what issue might be i haven't dropped my phone recently and it was working fine earlier one time it got on updated itself even though there was less memory and after that i can't get it on.
Please help to troubleshoot the problem and i also have important stuff i haven't backed up so formatting is not the option unless i am able to backup. My phone doesnt even connect to my win 7 pc, it says unable to recognise device i have also tried to deal with that but nothing happened but anyway this isn't the main problem, i’m just telling i might not be able to connect it to my pc too.
Any help will be appreciated .
 
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