perf(backend): cache YouTube searches and shrink the thumbnail URLs

Repeated searches and song lookups now reuse a single-flight memory cache, so
concurrent or debounced queries hit YouTube once instead of on every keystroke.
Thumbnails are resolved at a configurable size instead of the largest variant,
which cuts the image payload the browser pulls from Google.

backend/Musify.Infrastructure/Services/YouTubeThumbnail.cs

@@ -0,0 +1,52 @@
+namespace Musify.Infrastructure.Services
+{
+    public static class YouTubeThumbnail
+    {
+        public static string WithSize(string url, int size)
+        {
+            if (string.IsNullOrEmpty(url))
+                return string.Empty;
+
+            if (url.Contains("googleusercontent.com", StringComparison.OrdinalIgnoreCase))
+                return ResizeGoogleUserContent(url, size);
+
+            if (url.Contains("ytimg.com", StringComparison.OrdinalIgnoreCase))
+                return ResizeYtImg(url, size);
+
+            return url;
+        }
+
+        private static string ResizeGoogleUserContent(string url, int size)
+        {
+            var separator = url.LastIndexOf('=');
+            var baseUrl = separator > 0 && IsSizeSuffix(url.AsSpan(separator + 1))
+                ? url[..separator]
+                : url;
+
+            return $"{baseUrl}=w{size}-h{size}-l90-rj";
+        }
+
+        private static string ResizeYtImg(string url, int size)
+        {
+            var queryStart = url.IndexOf('?');
+            var path = queryStart >= 0 ? url[..queryStart] : url;
+
+            var lastSlash = path.LastIndexOf('/');
+            if (lastSlash < 0)
+                return path;
+
+            return $"{path[..lastSlash]}/{YtImgFileName(size)}";
+        }
+
+        private static string YtImgFileName(int size) => size switch
+        {
+            <= 120 => "default.jpg",
+            <= 320 => "mqdefault.jpg",
+            <= 480 => "hqdefault.jpg",
+            _ => "maxresdefault.jpg"
+        };
+
+        private static bool IsSizeSuffix(ReadOnlySpan<char> suffix) =>
+            suffix.Length > 1 && (suffix[0] is 'w' or 's' or 'h') && char.IsDigit(suffix[1]);
+    }
+}