EXCURSION FROM ASSIGNED ALT. TCASSII INDUCED.

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PROCEEDING DIRECT TO STONEWALL VOR HDG 120 DEGS AT FL370. E/W LINE OF TSTMS VERY ACTIVE IN CLOUD LIGHTNING TOPS TO FL400. WE REQUESTED AND WERE GIVEN E DEV TO AVOID A BUILD-UP AT OUR 12-1 O'CLOCK POS. ANOTHER CELL AT 9-10 O'CLOCK WAS ALSO A FACTOR. WHEN STEADY ON OUR DEV HDG; CTR CALLED TFC AT OUR ALT AND 2 O'CLOCK POS. AND GAVE US AN IMMEDIATE L TURN OF 45 DEGS. WE WERE AWARE OF THIS ACFT ON TCASII AND COULD SEE HIM VISUALLY AT ALL TIMES. HE WAS DEVIATING AROUND THE SAME CELL AND WE HAD BEEN WATCHING HIM. HE WAS COMING FROM THE W AND EVIDENTLY BEGAN DEVIATING MORE EASTERLY THAN CTR ANTICIPATED CAUSING A CONFLICT. WHILE MAKING THE 45 DEG L TURN; CTR ASKED FOR MORE TURN TO T HE L. TCASII CALLED 'TFC' AND THEN A DSND COMMAND FOLLOWED ALMOST IMMEDIATELY BY A 'CLB CLB' COMMAND. AS THE L TURN INCREASED. SINCE WE WERE VISUAL AND WATCHING THE OTHER ACFT AND AWARE OF OUR CO- ALT WE HAD BEGUN TO CLB IN THE TURN AS THE INITIAL CLB COMMAND CAME OUT TO INSURE SEPARATION. WE CLBED TO FL377 AND ADVISED CTR OF TCASII DEV. THE EBOUND ACFT REMAINED IN SIGHT UNTIL PASSING BELOW AND BEHIND. WE RPTED THE DEV AND RETURNED TO FL370.