DURING CLIMBOUT; ACR LGT NEARLY COLLIDED WITH VFR UNK-SMA.

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I WAS WORKING W ARR/DEP COMBINED WITH MODERATE TFC. I WAS TRYING TO ESTABLISH RADIO CONTACT WITH AN ACFT THAT HAD BEEN HANDED OFF. AS I WAS VISUALLY SCANNING MY TFC; I NOTICED A VFR SYMBOL APPEAR ABOUT 12 MI SW OF THE ARPT HDG N AND I HAD LGT X RWY HDG OFF RWY 25L CLBING TO 11000'. INITIALLY THE VFR HAD NO ALT READOUT. AS THE TARGETS WERE GETTING CLOSER THE VFR SHOWED AN ALT OF 4400' WITH LGT X CLBING OUT OF 3700'. I ISSUED A SAFETY ALERT TO LGT X AND ADVISED HIM TO MAINTAIN 4000'. I THEN TURNED ACR TO HDG 240 DEGS. THIS OCCURRED IN THE SPAN OF 3-4 SWEEPS OF THE RADAR. I DON'T THINK MUCH MORE COULD HAVE BEEN DONE. THE VFR SEEM TO COME OUT OF A BLIND SPOT OR COULD HAVE BEEN IN THE VFR PRACTICE AREA THAT STARTS APPROX WHERE THE INCIDENT OCCURRED AND MAY HAVE BEEN IN A MAX CLB. CALLBACK CONVERSATION WITH RPTR ACN 91508 REVEALED THE FOLLOWING: THE ACR LGT FLT CREW FILED AN OFFICIAL NMAC RPT WITH THEIR COMPANY. RPTR WAS NOT SURE HOW CLOSE THE 2 ACFT DID COME TO EACH OTHER; BUT DID HEAR FROM THE CAPT OF THE LGT THAT THE F/A HAD RPTED THE SMA AS DIVING FROM JUST ABOVE THE LGT TO OUT OF SIGHT BELOW. RPTR CHARACTERIZED THE PROX AS 'VERY CLOSE.' RPTR WAS GIVEN A LETTER OF APPRECIATION FROM HIS FAC FOR HIS TIMELY TFC ALERT. THE NTAP OF THE INCIDENT SHOWED THAT THE LGT DID STOP ITS CLB BELOW THE SMA. IF THE SMA HAD NOT DIVED; THE SEP WOULD HAVE BEEN BETTER THAN IT WAS. SUPPLEMENTAL INFO FROM ACN 91534: RECEIVED TFC ADVISORY FROM DEP CTL OF TFC AT 9 O'CLOCK; 1 MI NBOUND AT 4400'; AND TOLD TO MAINTAIN 4000'. WE WERE CLBING THROUGH 4000' AT TIME AND I PUSHED NOSE OVER BUT MOMENTUM CARRIED US TO 4250' BEFORE DSNDING BACK TO 4000'. WAS THEN TOLD THAT TFC HAD MADE AN ABRUPT DSCNT TO 3500'. WAS ISSUED CLRNC TO CLB TO 9000' WITH NO FURTHER INCIDENTS. F/A IN LEFT REAR CABIN OBSERVED A SINGLE ENG; LOW WING; WHITE WITH RED TRIM ACFT INVERTED AND DIVING. HE ESTIMATED THE ACFT TO BE WITHIN 1/4 MI OF US. I APPRECIATE THE DEP CTLR ISSUING THE ADVISORY; BUT THINK HIS CLRNC FOR US TO MAINTAIN 4000' AT THAT POINT WAS AN ERROR. WE WERE CLBING THROUGH 4000' AT 1500 FPM AND THAT ACTION PUT US ALMOST LEVEL WITH THE TARGET'S ALT BEFORE WE COULD REVERSE THE ASCENT AND LEVEL OFF.