python3, 46 bytes

Result

WA

Code [DL]

i=1
while 1:i=input(1^(int(i+"001",2)**.5%1>0)

stdin

00111011
00110111
11111110
01110110
01101111
11000001
00101001
10110111
01011011
00110111
11010101
00100000
01011010
10101011
11101110
11001011
00100011
00000011
01110011
00000000
10000110
10010111
10011111
00000001
11011110
10101000
01101101
10001000
00100100
10110010
00101101
10011001
11111101
00001000
11100111
01000110
01101001
10100101
00000110
01000010
00001010
00001111
01111000
00010101
00001111
10010010
00011100
11010010
01001110
10111110

stdout

stderr

  File "/volume/CODE", line 3
    
                                              ^
SyntaxError: unexpected EOF while parsing

strace