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So the phone can only boot to "ClockworkMod Recovery" due to unknown reason.
Is there a way to transfer files between my phone's internal memory and my computer? There is no SD card slot on the phone.
I don't have any other ROM in my phone which means I can't flash anything but the official ROM (through download mode) which I think will remove all the files.
This suggestion assumes familiarity with connecting to the phone via ADB
I had a similar situation on my previous phone. I was able to pull the important stuff over ADB when the phone was in recovery. In my case I had to mount some of the filesystems to access everything I wanted to retrieve.
I'm not sure if it is dependent on the phone whether the ADB service is available in recovery. On mine it was available in both recovery and download mode.
If ADB on your computer recognizes and can connect to your phone you should be able to retrieve and save any files you need.
If you aren't familiar with ADB this is a good place to start: http://www.xda-developers.com/android/all-in-one-guide-to-adb/
If you need any help with it let me know, ADB is a whole other topic so keeping it simple here.
On my CWM I have "mount USB storage" option under the menu "mounts and storage".
This video shows you how to do it with links to programs you need to download.
