ACR X FALSE TCASII TA RA CLB CAUSED LTSS FROM ACR Y. EVASIVE ACTION TAKEN.

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ACR X DURING CRUISE TCASII INDICATED A DISPLAY AT 10 O'CLOCK AND 10 MI INDICATING AN ACFT AT FL306. DISPLAY TURNED TO RA AT APPROX 10 SECONDS LATER. TCASII AVOIDANCE MANEUVER WAS INITIATED WITH AN INITIAL 'CLB.' AT THE TIME I XMITTED THAT ACR X WAS FOLLOWING A TCASII RA AND WAS CLBING. MANEUVER WAS FOLLOWED BY ANOTHER RA TO 'DSND.' ANOTHER ACFT WAS NOTED TO BE AT APPROX 1 O'CLOCK AND 2 MI INDICATING FL330. PREVIOUS FL306 RA RETURNED TO A TA INDICATING AT ABOUT 5 O'CLOCK POS. ZHU WAS QUESTIONED 3 TIMES IF THEY HAD ANY TFC AT FL310 OR NEAR OUR ALT AND POS. THEY REPLIED THAT THEY HAD NO TFC IN REF TO OUR POS. SUPPLEMENTAL INFO FROM ACN 299814: DURING CRUISE; TCASII INDICATED TFC AT 10 O'CLOCK AND 10 MI INDICATING AN ALT OF FL306. WE RECEIVED A TA FOLLOWED IMMEDIATELY BY AN RA AND A COMMAND TO CLB; WHICH WE DID. WE CLBED TO FL316 WHICH SATISFIED THE TCASII; BUT RECEIVED ANOTHER RA TO DSND DUE TO ANOTHER ACFT AT FL330; WHICH WE HAD VISUAL CONTACT. THE CAPT RPTED TO CTR OUR INTENTIONS TO CLB DURING OUR CLBOUT OF FL310. THE TCASII INDICATED THE FL306 TFC PASSED DIRECTLY BELOW OF US AND OUT TO OUR 5 O'CLOCK POS. AFTER QUESTIONING ZHU ABOUT THE FL306 TFC; HE INDICATED THAT HE DID NOT HAVE A TARGET AT FL306; ONLY THE FL330 AND US.