PA-34 Flight Instructor reported an NMAC event during cruise as ATC was slow in providing vectors; that resulted in both aircraft executing evasive maneuvers.

Narrative

[I] was talking to ZZZ ATC in [the] vicinity of ZZZ1. ATC advised traffic 9 o'clock; 2 miles; same altitude. Saw traffic on ADS-B but did not have visual confirmation. Advised ATC; no contact. ATC then directed me to climb to 3;500 - did not say immediate climb; just climb to 3;500 ft. Initiated climb. Still scanned for traffic and no traffic observed. While in climb; ATC - different voice - said; 'STOP climb; descend.' I started a descent. ATC stated; 'STOP descent; it appears traffic is breaking away.' At that point; I noticed aircraft on left wing; about 200 ft. from me in a steep descending right turn to pass behind me. I identified the aircraft as a low-wing but that is all I could identify; focused on flying. I asked ATC to mark the location and that I wanted to file a report for a near-miss. Flight continued to ZZZ3 and landed without incident. I subsequently phoned the radar facility and spoke to the Supervisor about the situation. In a subsequent email; I received info from the pilot of the other aircraft; whom I know personally. His account for it is below.That was me in Aircraft Y between ZZZ2 and ZZZ1 today on the 170 heading. It was a real demonstration in how onboard traffic technology can lead us down a dangerous path. I [was] watching you on ADS-B and opted to climb to increase the separation but it looked like you had the same thought. That was followed by a rapid descent which you mirrored as well. Interesting how this incredible technology can lead into unintended consequences.My takeaway is to make an early heading change rather than altitude whenever possible. In hindsight; I probably should have taken a southwest heading initially. Also; 98% of the time I use flight following. However; today I was on a 1200 code. The aircraft was probably blocked by the left windshield strut - I was flying right seat - as I never observed the traffic until the last minute. Perhaps if ATC would have issued something like 'BREAK/BREAK Aircraft X IMMEDIATE RIGHT CLIMB' it would have conveyed more urgency. Better yet; ATC should have provided a vector sooner. ATC Supervisor jumped in and provided the needed vectors; I understand primary ATC controller was training a new person and that may have caused some distractions. Fortunately; [the] other pilot; ATP retired pilot with 30;000+ hours; saw me and avoided. [I recommend a] joint phone call with ATC [and] both pilots to write up an article so others can learn from it.