NMAC OCCURS WHEN DEP CTLR FAILS TO PROVIDE SEPARATION BTWN A DEP EMJ AND A XING C172 S OF JAN; MS.

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DEPARTED RWY 16R. ASSIGNED RWY HDG TO 3000 FT. SHORTLY AFTER TKOF; PRIOR TO ATC TA; SAW TARGET ON TCASII INSIDE 6 MI RANGE; 2 O'CLOCK POS AND AT LEAST 1000 FT ABOVE. ATC THEN ADVISED C172 VFR TFC AT SAME POS. TFC WAS NOT IN SIGHT. ATC THEN ISSUED R TURN 270 DEGS; CLB TO 10000 FT. I INITIATED CLB AND TURN BUT STILL NOT SEEING TFC; THAT I WAS TURNING TOWARD AND CLBING INTO; STOPPED TURN ON 180 DEG HDG. REDUCED CLB APCHING 3000 FT. ADVISED FO TO INFORM ATC AS SUCH. ATC ADVISED MAINTAIN 3000 FT. AT THIS TIME; ACQUIRED TFC VISUALLY 1-2 O'CLOCK POS AND ABOVE OUR ALT. TFC PASSED CLOSE OFF OUR L AND ABOVE US. ATC ADVISED TO MAINTAIN VISUAL SEPARATION AND CLB TO 10000 FT. SIMULTANEOUSLY; WITH VISUAL ACQUISITION; RECEIVED TCASII. BOTH FO AND I CONFIRMED TFC WAS ABOVE US; AND CLB WOULD HAVE WORSENED SIT. NO FURTHER ACTION NOR INSTRUCTION FROM ATC RELATIVE TO TFC GIVEN. PROCEEDED NORMALLY. SUPPLEMENTAL INFO FROM ACN 549380: A C172 WAS AT OUR 1-2 O'CLOCK POS AT 3500 FT. OUR INSTRUCTIONS WERE GIVEN TO US AS WE WERE PASSING THROUGH 3000 FT. WE DSNDED BACK DOWN TO 3000 FT AND ASKED FOR A 180 DEG HDG. DEP STATED NEGATIVE AND TO CONTINUE TURN AND CLB. WE PASSED UNDER THE C172 BY APPROX 500 FT.