I don’t know if this will help in proving the Collatz Conjecture, but I’ve recently found it interesting that some Pythagorean triples (a,b,c) so a^2+b^2=c^2 like (3,4,5) have the Collatz sequences 5→16→8→4 and 3→10→5 and 3→10→5→16→8→4 and for the triple (5,12,13) the sequence has 12→6→3→10→5 and 13→40→20→10→5.
Some Pythagorean triples like (11,60,61) and (13,84,85) don’t work. In the specific Collatz relationship of a=3c+1/2^k, k=13 and 19 don’t work but k=2,3,5,7,9,11,15,17 all have solutions.
the cases for a=3c+1/2^k can get enormous with this example k=7 where a=m^2-n^2, b=2mn, c=m^2+n^2 (m,n)=(3739409260606175355129, 3652770257191091034598) and (2^k-3)m^2-(2^k+3)n^2=1 is satisfied, confirming this case from the Pell-type equations.
There are infinitely many Pythagorean triples that go from one value to the other through the Collatz sequence, but they get rarer as the numbers get bigger.