Add server-side folder picker

New GET /api/browse endpoint lists subdirectories at any path.
UI gets a folder icon button next to each path input that opens
a browsable directory tree modal. Escape or Cancel closes it,
clicking a folder navigates into it, Select confirms the choice.

Co-Authored-By: Claude Sonnet 4.6 <noreply@anthropic.com>

.claude/launch.json

@@ -0,0 +1,11 @@
+{
+  "version": "0.0.1",
+  "configurations": [
+    {
+      "name": "dup-finder-api",
+      "runtimeExecutable": "uvicorn",
+      "runtimeArgs": ["main:app", "--host", "0.0.0.0", "--port", "8000", "--reload"],
+      "port": 8000
+    }
+  ]
+}