  The Chabauty--Kim method and its refined variant by Betts and Dogra aim to
cut out the $S$-integral points $X(\mathbb{Z}_S)$ on a curve inside the
$p$-adic points $X(\mathbb{Z}_p)$ by producing enough Coleman functions
vanishing on them. We derive new functions in the case of the thrice-punctured
line when $S$ contains two primes. We describe an algorithm for computing
refined Chabauty--Kim loci and verify Kim's conjecture over $\mathbb{Z}[1/6]$
for all choices of auxiliary prime $p < 10{,}000$.
