ACFT ASSIGNED A NEW RWY BY APCH CTL. AFTER ESTABLISHED ON LOC; THE ACFT WAS INSTRUCTED TO GO AROUND AS IT WAS APCHING THE WRONG RWY.

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DURING THE STAR PORTION OF THE APCH; WAS INFORMED BY SOCAL APCH TO EXPECT RWY 24L AT LAX. UPON LEAVING SMO VOR; WAS GIVEN VECTORS FOR RWY 24L. THE APCH CTLR REQUESTED THAT WE ADVISE HIM WHEN WE HAVE THE RWY IN SIGHT. WE INFORMED THE CTLR THAT THERE WAS A CLOUD LAYER BTWN US AND THE ARPT. THE APCH CTLR THEN ISSUED INSTRUCTIONS TO TURN R TO INTERCEPT THE ILS RWY 25R LOC; DSND AND MAINTAIN 2300 FT. THIS WAS A RWY CHANGE FROM OUR INITIAL EXPECTED RWY OF 24L; WHICH WE WERE SET UP FOR; TO RWY 25R WHICH NECESSITATED A LAST MIN CHANGE OF APCH PLATES. THE RWY WAS STILL OBSCURED BY A CLOUD LAYER. UPON BECOMING ESTABLISHED ON THE ILS RWY 25R LOC; THE APCH CTLR STATED THAT WE WERE ON FINAL TO THE WRONG RWY WITH A PLANE ON THE RWY AND TO IMMEDIATELY GO AROUND. WE PERFORMED A GAR AS INSTRUCTED; WAS VECTORED BACK AROUND AND WAS CLRED FOR AN ILS RWY 25R APCH; SIDESTEP TO RWY 25L. LANDED RWY 25L AND WHILE TAXIING TO FBO WAS DIRECTED TO CALL THE TWR PHONE. TWR PERSONNEL WERE UNDER THE IMPRESSION THAT WE WERE NOT IN COM WITH ANYONE ON THE APCH TO RWY 25R PRIOR TO GAR. INFORMED THE TWR THAT WE WERE CONTINUALLY IN COM WITH THE APCH CTLR AND HAD NEVER BEEN SWITCHED OVER TO TWR. AT THE TIME OF ARR INTO LAX AREA; THERE WAS CONSIDERABLE INCOMING/DEPARTING AIR TFC WITH A BUSY WORKLOAD FOR ALL CTLRS. SUPPLEMENTAL INFO FROM ACN 431391: I BELIEVE THE PROB AROSE FROM HVY TFC AND CTLR OVERLOAD.