feat(cli): add page icon frontmatter support

apps/cli/sync.ts

@@ -5,8 +5,8 @@
  * - New problems get a full page (frontmatter, badge, warning, examples,
  *   constraints, solution) in the gold-standard format.
  * - Already-ported pages only get their script-owned parts regenerated — the
- *   `<Badge>` line and the `## Solution` section — so hand-curated prose is
+ *   frontmatter `icon:` (`PAGE_ICON`), the `<Badge>` line and the
- *   never touched.
+ *   `## Solution` section — so hand-curated prose is never touched.
  * - When a problem is solved in both JavaScript and Python the solution
  *   renders as a <CodeGroup> (Python first); single-language solutions
  *   render as a plain fence.
@@ -417,6 +417,9 @@ function warningText(h: Header): string {
   );
 }
 
+/** Sidebar/page icon for every problem page (Lucide, resolved by Blume). */
+const PAGE_ICON = "line-dot-left-horizontal";
+
 function badgeLine(sol: Solution): string {
   return `<Badge variant="accent">${sol.category}</Badge>`;
 }
@@ -431,6 +434,7 @@ function renderPage(sol: Solution): string {
     "---",
     `title: '${h.num}. ${h.title.replaceAll("'", "''")}'`,
     /[:#]/.test(description) ? `description: '${description.replaceAll("'", "''")}'` : `description: ${description}`,
+    `icon: ${PAGE_ICON}`,
     "sidebar:",
     `  label: '${h.title.replaceAll("'", "''")}'`,
     `  badge: '${h.difficulty}'`,
@@ -508,6 +512,17 @@ function ensureLabel(page: string, h: Header): string {
   return page.replace(/^sidebar:$/m, `sidebar:\n${label}`);
 }
 
+/** Set the frontmatter's top-level `icon:` to `PAGE_ICON`, inserting it before `sidebar:`. */
+function ensureIcon(page: string): string {
+  const fm = page.match(/^---\n[\s\S]*?\n---/);
+  if (!fm) return page;
+  const line = `icon: ${PAGE_ICON}`;
+  const next = /^icon: .*$/m.test(fm[0])
+    ? fm[0].replace(/^icon: .*$/m, line)
+    : fm[0].replace(/^sidebar:$/m, `${line}\nsidebar:`);
+  return next + page.slice(fm[0].length);
+}
+
 /**
  * Offset just past the script-owned `## Solution` section: the next top-level
  * heading that is not inside a code fence, or end of file. Sections a human
@@ -564,7 +579,7 @@ for (const sol of ordered) {
     }
     const tail = page.slice(solutionEnd(page, at));
     const body = page.slice(0, at) + solutionSection(sol) + (tail ? `\n${tail}` : "");
-    const next = ensureLabel(body, sol.header).replace(/^<Badge variant="accent">.*<\/Badge>$/m, badgeLine(sol));
+    const next = ensureIcon(ensureLabel(body, sol.header)).replace(/^<Badge variant="accent">.*<\/Badge>$/m, badgeLine(sol));
     const actions: string[] = [];
     if (existing !== name) {
       await rename(join(DOCS, existing), join(DOCS, name));