riffing off the Collatz process

Definition 1. Let X be an n-tuple of positive integers. Then the k-th term of X is said to be a triangle-admissible member of X fif the terms of X up to and including the k-th term can be partitioned into three sets A, B, and C such that sumA, sumB, and sumC satisfy the triangle inequality.
Definition 2. A given positive integer is said to be Collatz-compatible fif its associated Collatz process reaches 1 (i.e., if its total stopping time is finite).
Definition 3. For each Collatz-compatible positive integer n, let X be the m-tuple of the terms of the Collatz process for n, with m being its total stopping time, and let p be the number of the terms of X that are triangle-admissible. Then f(n) = p/m.
QUERY: Given the sequence {n} of Collatz-compatible positive integers, what does the sequence {f(n)} look like?
QUERY 2. Among the triangle-admissible terms, are there any that admit a Pythagorean triple?
keywords: Mathematics, open problem
(posted to CI on 19.Jul.2019)