haskell, 106 bytes

Result

AC

Duration

856ms

Code [DL]

main=interact$concatMap f.lines;f x=show(0!(x++"__"))++" ";n!(x:s:p)|x=='|'=n+1!(s:p)|s=='|'=(n+1)!p|0<1=n

stdin

||_|__||__
|||___|_|_
||_|||_|||
|__|_|___|
|_____|___
||__|||_||
|||||___||
|___|_||_|
||__||___|
|_||___|_|
||_|______
|_|||___||
|__||_|__|
|_|___||_|
||||||_|||
||__|_|_|_
||_|___|||
|_____|_||
||____||||
|||____|||
||_|||||_|
|||||||__|
||__|___|_
|_||||||||
|__|_|||__
||||_|____
||_|__|___
|_______||
||__||||_|
||||||__||

stdout

3 3 8 1 1 2 5 1 2 3 3 4 1 2 9 2 3 1 2 3 8 7 2 9 1 5 3 1 2 6 

stderr

/volume/CODE:1:61: warning: [-Wmissing-space-after-bang]
    Did you forget to enable BangPatterns?
    If you mean to bind (!) then perhaps you want
    to add a space after the bang for clarity.
  |
1 | main=interact$concatMap f.lines;f x=show(0!(x++"__"))++" ";n!(x:s:p)|x=='|'=n+1!(s:p)|s=='|'=(n+1)!p|0<1=n
  |                                                             ^^^^^^^^

strace

10    execve("/bin/sh", ["/bin/sh", "/root/script", "/volume/CODE"], 0x7ffd65d76d50 /* 7 vars */) = 0
12    execve("/usr/bin/runghc", ["/usr/bin/runghc", "/volume/CODE"], 0x5570060fdd40 /* 7 vars */) = 0
11    execve("/bin/cat", ["cat", "-"], 0x5570060fdd30 /* 7 vars */ <unfinished ...>
12    execve("/usr/lib/ghc-8.8.3/bin/runghc", ["/usr/lib/ghc-8.8.3/bin/runghc", "-f", "/usr/bin/ghc-8.8.3", "/volume/CODE"], 0x558813328168 /* 7 vars */ <unfinished ...>
11    <... execve resumed>)             = 0
12    <... execve resumed>)             = 0
11    +++ exited with 0 +++
10    --- SIGCHLD {si_signo=SIGCHLD, si_code=CLD_EXITED, si_pid=11, si_uid=0, si_status=0, si_utime=0, si_stime=0} ---
12    execve("/usr/bin/ghc-8.8.3", ["/usr/bin/ghc-8.8.3", "-ignore-dot-ghci", "-x", "hs", "-e", ":set prog \"/volume/CODE\"", "-e", ":main []", "/volume/CODE"], 0x7ffd025caf20 /* 7 vars */) = 0
12    execve("/usr/lib/ghc-8.8.3/bin/ghc", ["/usr/lib/ghc-8.8.3/bin/ghc", "-B/usr/lib/ghc-8.8.3", "-ignore-dot-ghci", "-x", "hs", "-e", ":set prog \"/volume/CODE\"", "-e", ":main []", "/volume/CODE"], 0x558ffc113200 /* 7 vars */) = 0
17    execve("/bin/gcc", ["gcc", "-DTABLES_NEXT_TO_CODE", "-B/usr/lib/ghc-8.8.3/integer-gmp"..., "--print-file-name", "libgmp.so"], 0x7ffcb6c97fe0 /* 7 vars */) = -1 ENOENT (No such file or directory)
17    execve("/usr/local/sbin/gcc", ["gcc", "-DTABLES_NEXT_TO_CODE", "-B/usr/lib/ghc-8.8.3/integer-gmp"..., "--print-file-name", "libgmp.so"], 0x7ffcb6c97fe0 /* 7 vars */) = -1 ENOENT (No such file or directory)
17    execve("/usr/local/bin/gcc", ["gcc", "-DTABLES_NEXT_TO_CODE", "-B/usr/lib/ghc-8.8.3/integer-gmp"..., "--print-file-name", "libgmp.so"], 0x7ffcb6c97fe0 /* 7 vars */) = -1 ENOENT (No such file or directory)
17    execve("/usr/sbin/gcc", ["gcc", "-DTABLES_NEXT_TO_CODE", "-B/usr/lib/ghc-8.8.3/integer-gmp"..., "--print-file-name", "libgmp.so"], 0x7ffcb6c97fe0 /* 7 vars */) = -1 ENOENT (No such file or directory)
17    execve("/usr/bin/gcc", ["gcc", "-DTABLES_NEXT_TO_CODE", "-B/usr/lib/ghc-8.8.3/integer-gmp"..., "--print-file-name", "libgmp.so"], 0x7ffcb6c97fe0 /* 7 vars */) = 0
17    +++ exited with 0 +++
12    --- SIGCHLD {si_signo=SIGCHLD, si_code=CLD_EXITED, si_pid=17, si_uid=0, si_status=0, si_utime=0, si_stime=0} ---
14    +++ exited with 0 +++
15    +++ exited with 0 +++
16    +++ exited with 0 +++
18    +++ exited with 0 +++
13    +++ exited with 0 +++
12    +++ exited with 0 +++
10    --- SIGCHLD {si_signo=SIGCHLD, si_code=CLD_EXITED, si_pid=12, si_uid=0, si_status=0, si_utime=20, si_stime=4} ---
10    +++ exited with 0 +++