A LOW TIME PVT PLT EXPERIENCES A POTENTIAL CONFLICT WHEN TURNING BASE TOO SOON AFTER SIGHTING THE WRONG TFC ON A POINTOUT FOR RWY 13L AT BFI; WA.

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I PRACTICED LNDGS AT BFI. THE ARPT HAS 2 PARALLEL RWYS (10001 FT BY 200 FT AND 3710 FT BY 100 FT). TPA'S ARE 800 FT AND 1000 FT RESPECTIVELY. THE LONGER ONE IS CTLED ON 120.6 MHZ AND THE SHORTER ONE ON 118.3 MHZ. I WAS FLYING THE SHORTER RWY 13L/31R PATTERN WITH SOUTHERN FLOW AT THE TIME. I WAS APCHING MIDFIELD ON THE L DOWNWIND TO RWY 13L WHEN THE TWR ADVISED ME TO FOLLOW CESSNA ON 1 MI FINAL AND CLRED ME FOR THE TOUCH-AND-GO. I ACKNOWLEDGED. A COUPLE OF SECONDS LATER; I SPOTTED A CESSNA ON APPROX 1 MI FINAL AND ADVISED THE TWR THAT I HAD THE TFC IN SIGHT. I FLEW AN EXTENDED DOWNWIND LEG UNTIL THIS TFC WAS ABEAM MY L WING AND TURNED BASE. ONLY AT THIS MOMENT DID IT BECOME APPARENT THAT THE TFC I SAW WAS LNDG ON RWY 13R. NOT BEING ABLE TO FIND THE TFC I WAS INSTRUCTED TO FOLLOW; I ANNOUNCED MYSELF ON L BASE TO RWY 13L. THE TWR ADVISED ME TO GO AROUND BECAUSE; AS THE CTLR PUT IT '...YOU CUT OFF YOUR TFC.' I ACKNOWLEDGED AND FOLLOWED THE INSTRUCTION NEVER SEEING 'MY TFC.' IT WAS A BRIGHT; CLR; SUNNY DAY WITH A LITTLE BIT OF HAZE. AT XA30; THE SUN WAS CLOSE TO BEING ABEAM MY L WING WHILE ON DOWNWIND. THE TFC I WAS LOOKING FOR WAS VERY DIFFICULT TO SPOT ON THE CITY STREETS BACKGND. I WAS SURPRISED I WAS ABLE TO 'FIND IT.' THE RWY 13R THRESHOLD 'PROTRUDES' 1/4 MI FURTHER NW THAN RWY 13L. TFC LNDG ON RWY 13R LOOKED APPROX 1 MI FROM RWY 13L THRESHOLD. FROM THE POS I WAS IN WHEN FIRST ADVISED TO FOLLOW ANOTHER TFC; IT WAS VERY DIFFICULT TO JUDGE ON WHICH RWY THE TFC I SPOTTED WAS LNDG. I JUST 'SAW/ASSUMED' IT WAS RWY 13L. IT WAS A MISTAKE. AS FOR MYSELF; I WILL NEVER ASSUME ANYTHING IN A SIT LIKE THIS ONE. I WILL FLY THE EXTENDED DOWNWIND LEG UNTIL I ACTUALLY SEE THE TFC IN FRONT OF ME CLOSING IN ON THE THRESHOLD OF THE RWY I INTEND TO LAND ON OR WILL KEEP ASKING THE TWR ABOUT ITS POS. THE SMALL 'RISK' OF HEARING 'XYZ YOU CAN START YOUR BASE NOW' IS MUCH BETTER THAN WHAT I HEARD. AS FOR THE CTLRS IN AN ARPT WITH THIS KIND OF A LAYOUT; WITH MULTIPLE AIRPLANES LNDG ON BOTH RWYS; WITH THAT TYPE OF 'LIGHTING' CONDITIONS; AND RADAR DATA AVAILABLE; I THINK IT IS DESIRABLE TO CALL PLT'S BASE WHEN THERE IS NO CONFLICT ANY LONGER. SO; INSTEAD OF 'XYZ FOLLOW ABC ON 1 MI FINAL CLR...' IT SHOULD BE SOMETHING LIKE 'XYZ FOLLOW ABC ON 1 MI FINAL; WE WILL CALL YOUR BASE' AND GIVE THE CLRNC AT THAT TIME.