When the first loop finds the first line you wants to process, it exits with the line still being in $_. Then the condition of the second while loop reads the next line to $_ thus discarding the first one. You should write

while (<LOG>) {
 ... processing log line ....
}
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to

do {
 ... processing log line ....
} while <LOG>;
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Because of the special way perl handles do {} while, the condition is evaluated only after the body has run first, so the body processed the line that's still in $_.