Step 1: Frame this around purine salvage. Cells reuse free purine bases instead of always making them from scratch, and the key salvage enzyme for hypoxanthine and guanine is $HGPRT$.
Step 2: When the X-linked $HPRT$ gene is mutated, $HGPRT$ activity is lost, and this is exactly the defect in Lesch-Nyhan syndrome, also called juvenile gout.
Step 3: The clinical fallout follows logically: unsalvaged bases are funneled into uric acid, producing hyperuricemia, while neurologic damage gives self-injury, dystonia and cognitive impairment.
Step 4: Reject the alternatives because they act on pyrimidines or NAD. Orotate and uracil phosphoribosyltransferases are pyrimidine enzymes, and quinolinate phosphoribosyltransferase feeds NAD synthesis, none tied to this syndrome.
\[\boxed{\text{Hypoxanthine-guanine phosphoribosyltransferase (HGPRT)}}\]