HDG TRACK DEV CREATES A POTENTIAL CONFLICT FOR ACR ACFT OVER MOUNT RUSHMORE.

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OVER MOUNT RUSHMORE; CAPT DECIDED TO DISCONNECT AUTOPLT AND BANK ACFT TO THE R TO GIVE PAX A GOOD VIEW OF MOUNT RUSHMORE. ACFT TURNED WELL INSIDE THE CLRED AIRWAY TRACK DIRECT TO RAP AND THEN RAP TO OBH VOR. ATC CALLED OUR FLT TO ASK OUR HDG SINCE WE WERE ABOUT TO CAUSE A CONFLICT WITH OTHER ACFT. I WAS OFF THE RADIO; ON THE PA; GIVING AN ANNOUNCEMENT ABOUT THE VIEW. CAPT MISSED THE RADIO CALLS. WE THEN BOTH HEARD ATC DSND ANOTHER ACFT 'IMMEDIATELY' FOR TFC. WE LATER FIGURED OUT THIS WAS BECAUSE OF OUR TURN. ATC CALLED US AGAIN; THIS TIME I RESPONDED. HE ASKED HDG. WE WERE TURNING BACK ON COURSE BY THAT TIME BUT HE TURNED US FURTHER N TO AVOID TFC. I WAS VERY SURPRISED TO SEE HOW FAR FROM OUR CLRED TRACK WE HAD STRAYED. MY ATTN WAS DIRECTED OUTSIDE AND WHEN I LOOKED BACK IN WE WERE 10 MI FROM TRACK. THE TURN JUST DIDN'T SEEM THAT GREAT TO ME SO CAUSED ME NO CONCERN. OBVIOUSLY I WAS MISTAKEN AS WE STRAYED A GREAT DEAL OFF OUR TRACK. THE SIT WAS MADE WORSE BY THE CAPT'S INATTN TO ATC'S RADIO CALLS. SUPPLEMENTAL INFO FROM ACN 276936: TFC CONFLICT AT 37000 FT. DSNDED OTHER ACFT TO 33000 FT. A PREVIOUS RADIO XMISSION NOT HEARD BUT ONE WAS BLOCKED BY OTHER ACFT MAKING AN INQUIRY.