Improve or verify the lower bound for the Schur number $S(6)$

The Schur number $S(r)$ is the largest integer $n$ such that the interval $\{1,2,\dots,n\}$ can be partitioned into $r$ sum-free sets — sets containing no solution to $x+y=z$ (with $x,y,z$ in the same part, $x,y$ not necessarily distinct). The values $S(1)=1,\;S(2)=4,\;S(3)=13,\;S(4)=44,\;S(5)=160$ are known; $S(5)=160$ was settled by a massive SAT computation (Heule, 2017). $S(6)$ is open: only a lower bound is known and no matching upper bound has been proved. Exhibit a partition of $\{1,\dots,n\}$ into $6$ sum-free sets for the largest $n$ you can, matching or exceeding the current best-known lower bound recorded in OEIS A045652 — or, if you can, close the gap entirely.

Acceptance. ADVANCES: a machine-checkable partition of $\{1,\dots,n\}$ into exactly 6 parts, given as an explicit color assignment, such that (a) every integer in $\{1,\dots,n\}$ receives exactly one color and (b) no part contains a triple $x+y=z$ — verified by a script that scans all pairs — where $n$ equals or exceeds the current best-known lower bound for $S(6)$ listed in OEIS A045652 at submission time. FULLY RESOLVES: either such a partition with $n$ strictly larger than the best known (a new record lower bound), or a verifiable proof (e.g. a SAT UNSAT certificate / DRAT proof that no 6-coloring of $\{1,\dots,m\}$ avoids a monochromatic Schur triple) establishing an upper bound $S(6)<m$, ideally pinning the exact value. Provide the coloring/certificate plus the verification script.