Unable to load OS X after installing BootCamp

I have unfortunately already edited the 'unallocated' space (appoximately a few hundred mb) and now i have no way to boot into my OS X. I think that is the reason why i am unable to boot into OS X.
The unallocated space just shows empty and i have allocate it to my ‘Windows' and now i can't boot it.
Before this problem exist, i will just press option when i hear the boot up sound and choose OS X. But this option is gone, all i can see it Windows.
I am not willing to delete my OS X because i have plenty of important files in there.
Is there any easy way to solve it?

I am using the Late 2013 15" Macbook Pro with Retina Dispaly

Yes, i can access Internet Recovery. I am unable to boot into Local Recovery though.
Holding Command+R/Command+Option+R only allows me to enter Internet Recovery. I can't seem to do anything there. I tried to Re-install Mac OS X but it states that 'The disk cannot be used to startup your computer'.
What else can i do?

-bash-3.2# diskutil list

/dev/disk0

#: TYPE NAME SIZE IDENTIFIER

0: GUID_partition_scheme *500.3 GB disk0

1: EFI EFI 209.7 MB disk0s1

2: Apple_CoreStorage 385.4 GB disk0s2

3: Microsoft Basic Data BOOTCAMP 114.7 GB disk0s3

/dev/disk1

#: TYPE NAME SIZE IDENTIFIER

0: Apple_partition_scheme *1.3 GB disk1

1: Apple_partition_map 30.7 KB disk1s1

2: Apple_HFS OS X Base System 1.3 GB disk1s2

/dev/disk2

#: TYPE NAME SIZE IDENTIFIER

0: Apple_HFS Macintosh HD *385.0 GB disk2

/dev/disk3

#: TYPE NAME SIZE IDENTIFIER

0: untitled *524.3 KB disk3

/dev/disk4

#: TYPE NAME SIZE IDENTIFIER

0: untitled *524.3 KB disk4

/dev/disk5

#: TYPE NAME SIZE IDENTIFIER

0: untitled *524.3 KB disk5

/dev/disk6

#: TYPE NAME SIZE IDENTIFIER

0: untitled *524.3 KB disk6

/dev/disk7

#: TYPE NAME SIZE IDENTIFIER

0: untitled *524.3 KB disk7

/dev/disk8

#: TYPE NAME SIZE IDENTIFIER

0: untitled *6.3 MB disk8

/dev/disk9

#: TYPE NAME SIZE IDENTIFIER

0: untitled *2.1 MB disk9

/dev/disk10

#: TYPE NAME SIZE IDENTIFIER

0: untitled *1.0 MB disk10

/dev/disk11

#: TYPE NAME SIZE IDENTIFIER

0: untitled *524.3 KB disk11

/dev/disk12

#: TYPE NAME SIZE IDENTIFIER

0: untitled *524.3 KB disk12

/dev/disk13

#: TYPE NAME SIZE IDENTIFIER

0: untitled *1.0 MB disk13

-bash-3.2# diskutil cs list

CoreStorage logical volume groups (1 found)

|

+-- Logical Volume Group B3C13BD8-2D05-4A88-8F4B-EF15C0474A95

=========================================================

Name: Macintosh HD

Status: Online

Size: 385371222016 B (385.4 GB)

Free Space: 35676160 B (35.7 MB)

|

+-< Physical Volume 4AA52C0C-57DF-444D-96F0-0AE7D06271D5

| ----------------------------------------------------

| Index: 0

| Disk: disk0s2

| Status: Online

| Size: 385371222016 B (385.4 GB)

|

+-> Logical Volume Family 331503DF-5806-488E-BB9E-E52A6F924B08

----------------------------------------------------------

Encryption Status: Unlocked

Encryption Type: None

Conversion Status: NoConversion

Conversion Direction: -none-

Has Encrypted Extents: No

Fully Secure: No

Passphrase Required: No

|

+-> Logical Volume 091E9D6C-AF21-4068-8845-F72953AF4B61

---------------------------------------------------

Disk: disk2

Status: Online

Size (Total): 384999997440 B (385.0 GB)

Conversion Progress: -none-

Revertible: Yes (no decryption required)

LV Name: Macintosh HD

Volume Name: Macintosh HD

Content Hint: Apple_HFS

-bash-3.2# gpt -vv -r show /dev/disk0

gpt show: /dev/disk0: mediasize=500277790720; sectorsize=512; blocks=977105060

gpt show: /dev/disk0: PMBR at sector 0

gpt show: /dev/disk0: Pri GPT at sector 1

gpt show: /dev/disk0: Sec GPT at sector 977105059

start size index contents

0 1 PMBR

1 1 Pri GPT header

2 32 Pri GPT table

34 6

40 409600 1 GPT part - C12A7328-F81F-11D2-BA4B-00A0C93EC93B

409640 752678168 2 GPT part - 53746F72-6167-11AA-AA11-00306543ECAC

753087808 704

753088512 224016384 3 GPT part - EBD0A0A2-B9E5-4433-87C0-68B6B72699C7

977104896 131

977105027 32 Sec GPT table

977105059 1 Sec GPT header

-bash-3.2# fdisk /dev/disk0

Disk: /dev/disk0 geometry: 60821/255/63 [977105060 sectors]

Signature: 0xAA55

Starting Ending

#: id cyl hd sec - cyl hd sec [ start - size]

------------------------------------------------------------------------

1: EE 1023 254 63 - 1023 254 63 [ 1 - 977105059] <Unknown ID>

2: 00 0 0 0 - 0 0 0 [ 0 - 0] unused

3: 00 0 0 0 - 0 0 0 [ 0 - 0] unused

4: 00 0 0 0 - 0 0 0 [ 0 - 0] unused

Sadly, i did not have any backup for OS X because i have re-allocate the size before and everything works fine for a few times and i didn't backup at all due to that but this time round i have no idea what is wrong.

My bootcamp (windows 10) is a free trial which i downloaded from online. I am willing to re-install my Windows since most of the things i have here are games, i am just not willing to lose my OS X stuffs because it contains my school work.

How much space do i need to install the version (10.11.6) i currently have?
Will i lose my OS X items?

By doing that, will i lose my Windows items too? (Just curious)

Thankyou for all your help! I managed to get back my OS X files but i have to delete my bootcamp as i do not have enough space to re-install OS X. Thank you so much for you help! Really appreciate it a lot!!

Nope, it is okay. Thankyou for your help!