TUG DRIVER PUSHES B737-700 ONTO ACTIVE TXWY AND CROSSES TAIL OVER RWY HOLD SHORT LINES.

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PUSHING FROM GATE; NO PUSH CALL REQUIRED. TUG DRIVER ASKED IF WE WERE GOING TO TAXI ON TXWY S. I SAID YES; AS THE ACTIVE WAS RWY 33L. I THOUGHT HE SAID HE WAS GOING TO PUSH BEYOND GATE X TO ALLOW INBOUND ACFT ACCESS TO GATE Y. AS THE PUSH PROGRESSED; IT BECAME APPARENT THAT WE WERE NOT GOING DEEPER INTO THE ALLEY; BUT TOWARDS TXWY S. WE DISMISSED THIS AS NO PROB BECAUSE WE CAN PUSH FAR ENOUGH TO LET ANOTHER ACFT INTO THE ALLEY AND STILL REMAIN CLR OF TXWY S. I BECAME DISTR BY MONITORING THE START AND FAILED TO NOTICE THAT WE WERE PUSHED WAY BACK ONTO TXWY S. WHEN THE TUG DRIVER ASKED FOR THE PARKING BRAKE; I NOTICED WE WERE ASTRIDE YELLOW BARS. THE TUG WAS CLRED OFF JUST AS I REALIZED THAT THOSE BARS WERE THE HOLD SHORT LINES FOR RWY 33L! SHORTLY AFTER THAT; GND CTL ASKED US TO MOVE FORWARD AS THEY HAD ALREADY SENT TFC AROUND. WHEN THE TUG ASKED IF WE WERE GOING TO TAXI ON TXWY S; I THINK HE WAS ASKING FOR PERMISSION TO PUSH US ONTO TXWY S. I SHOULD HAVE STOPPED THE PUSH ONCE THE TUG DEVIATED FROM THE EXPECTED PATH. I SHOULD HAVE NOT LET THE ENG START DISTRACT ME FROM KNOWING WHERE MY SHIP WAS GOING.