A TAXIING SF340 ENTERS RWY 23L AT CLE WITHOUT AUTH; THINKING IT WAS RWY 28.

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GND ISSUED TAXI INSTRUCTIONS FOR US TO GO TO RWY 23R; HOLD SHORT OF RWY 23L ON RWY 28. I WAS DOING THE BEFORE TKOF CHK AS THE CAPT WAS TAXIING. HE SAID HE THOUGHT HE HAD GONE PAST HIS TXWY; BUT WOULD TAKE THE NEXT ONE ONTO RWY 28. AS HE TAXIED ONTO THE RWY; I NOTICED AN ACFT ON SHORT FINAL; BUT WAS UNSURE AS TO WHAT RWY HE WAS USING. I QUESTIONED ATC AS TO OUR LOCATION. ATC TOLD THE ACFT TO GO AROUND. HE HAD INADVERTENTLY TAXIED ONTO RWY 23L THINKING IT WAS RWY 28. THE CAPT HAD TAKEN A L ON TXWY W AND NOT YET REACHED OUR ASSIGNED TXWY (I BELIEVE IT WAS TXWY U). HE SHOULD HAVE TURNED R ONTO TXWY W; THEN A SHORT L ONTO TXWY U(?) TO RWY 28; UP TO RWY 23L. NEITHER OF US REALIZED WE WERE AT THE WRONG POS UNTIL ATC TOLD THE LNDG ACFT TO GAR. SUPPLEMENTAL INFO FROM ACN 423338: TAXI INSTRUCTIONS WERE RWY 23L VIA TXWY W; TXWY U; RWY 28; SHORT OF RWY 23L. THAT AREA OF THE ARPT IS VERY CONFUSING AND I ENDED UP TURNING THE WRONG WAY ON TXWY W. I CLRED THE AREA FIRST; AND TAXIED ONTO WHAT I THOUGHT WAS RWY 28. SINCE TWR AND GND WERE COMBINED; I HEARD THE CTLR TALKING ABOUT AN ACFT GOING AROUND. I REALIZED I HAD TAXIED ONTO RWY 23L AND EXITED THE RWY QUICKLY AT TXWY L; A SHORT DISTANCE AWAY. WHEN I WAS CLR; THE CTLR GAVE ME THE SAME TAXI INSTRUCTIONS; TXWY W; TXWY U; RWY 28; SHORT OF RWY 23L AND TOLD ME ABOUT THE ACFT ON FINAL THAT HAD TO GO AROUND BECAUSE OF ME. THE CTLR DID NOT SAY ANYTHING MORE ABOUT THE INCIDENT. I DID CLR THE AREA AS BEST I COULD BEFORE TAXIING ONTO THE RWY; BUT BECAUSE IT WAS THE R SIDE; BEHIND ME; I COULD NOT SEE THE ACFT ON FINAL. OTHER CONTRIBUTING FACTORS WERE NIGHT; THE WX; AND THAT I KNEW I WOULD BE TAXIING ONTO RWY 28 -- SO SEEING RWY LIGHTS WAS WHAT I EXPECTED. I NEVER SAW THE ACFT ON FINAL.