python3, 43 bytes

Result

WA

Code [DL]

while 1:_=1^((1+8*int(input(_),2))**.5%1>0)

stdin

00110111
00100111
00100100
10011100
00111011
00000010
00111001
01000010
10011000
01111000
10000101
10000011
10110100
10101011
00110111
10010101
00000000
11110111
00001011
00011101
01001110
00011100
10111110
10111000
00000100
01011011
00001111
11100111
00010101
10100000
11100101
01101001
11111101
00000011
10001100
00001101
11010010
10001000
00000001
10111011
00010100
11111101
01100110
00001010
11110000
00101101
01000100
00000110
10011001
11110100

stdout

stderr

Traceback (most recent call last):
  File "/volume/CODE", line 1, in <module>
    while 1:_=1^((1+8*int(input(_),2))**.5%1>0)
NameError: name '_' is not defined

strace