Extend an open aliquot sequence of the Lehmer Five (276, 552, 564, 660, 966) to a new frontier, or resolve its fate (Guy UPINT §B6)

The aliquot sequence of $n$ iterates $s(n)=\sigma(n)-n$ (sum of proper divisors). Each sequence either terminates (reaches $0$/$1$ or a perfect number), enters a cycle (perfect, amicable, or sociable), or is unbounded — whether unbounded orbits occur is itself open (Catalan–Dickson vs. Guy–Selfridge). For the five smallest starting values whose fate is unknown, the 'Lehmer Five' $\{276, 552, 564, 660, 966\}$, extend a chosen sequence beyond its current published frontier index, or determine that it terminates or enters a cycle.

Acceptance. Choose one Lehmer-Five sequence and state its current frontier index and term (from FactorDB/rieselprime.de). ADVANCES: extend it by at least one new term beyond that frontier, supplying for each new step the full prime factorization of the term being iterated so that $s(n)=\sigma(n)-n$ is machine-checkable, plus a script that recomputes each $s$ from the factorizations. FULLY RESOLVES: a proof that the chosen sequence terminates or enters a cycle (exhibit the cycle) — currently believed intractable. Deliver the new terms, their predecessors' factorizations, and the recomputation script.