feat: add in-memory media info cache and YouTube search feature

- Add in-memory LRU cache (200 entries, 30 min TTL) to yt-dlp Client for
  GetMediaInfo and GetPlaylistInfo results, avoiding redundant yt-dlp calls
  for the same URL within a session
- Add YouTube (/search) and YouTube Music (/music) search with paginated
  results (5 per page), user selects a result to start the download flow
- Add SearchState type, new search.go handler, search results keyboard with
  < > navigation and New Search button
- Add Search button to main menu keyboard
- Update help text with new commands

internal/dl/ytdlp.go

@@ -10,6 +10,7 @@ import (
 	"os/exec"
 	"path/filepath"
 	"strings"
+	"sync"
 	"syscall"
 	"time"
 
@@ -20,6 +21,36 @@ import (
 type Client struct {
 	binaryPath  string
 	cookiesFile string
+	cache       map[string]*cachedMediaInfo
+	cacheMu     sync.Mutex
+}
+
+type cachedMediaInfo struct {
+	info *domain.MediaInfo
+	ts   time.Time
+}
+
+const cacheTTL = 30 * time.Minute
+const cacheMaxEntries = 200
+
+func (c *Client) getCached(url string) *domain.MediaInfo {
+	c.cacheMu.Lock()
+	defer c.cacheMu.Unlock()
+	entry, ok := c.cache[url]
+	if !ok || time.Since(entry.ts) > cacheTTL {
+		delete(c.cache, url)
+		return nil
+	}
+	return entry.info
+}
+
+func (c *Client) setCache(url string, info *domain.MediaInfo) {
+	c.cacheMu.Lock()
+	defer c.cacheMu.Unlock()
+	if len(c.cache) >= cacheMaxEntries {
+		c.cache = make(map[string]*cachedMediaInfo)
+	}
+	c.cache[url] = &cachedMediaInfo{info: info, ts: time.Now()}
 }
 
 // NewClient creates a new yt-dlp client. It checks for the binary.
@@ -29,7 +60,11 @@ func NewClient(cookiesFile string) (*Client, error) {
 		return nil, fmt.Errorf("yt-dlp not found in PATH: %w", err)
 	}
 	slog.Info("yt-dlp found", "path", path)
-	return &Client{binaryPath: path, cookiesFile: cookiesFile}, nil
+	return &Client{
+		binaryPath:  path,
+		cookiesFile: cookiesFile,
+		cache:       make(map[string]*cachedMediaInfo),
+	}, nil
 }
 
 type ytdlpFormat struct {
@@ -70,6 +105,11 @@ type ytdlpPlaylistEntry struct {
 
 // GetMediaInfo fetches format information for a given URL.
 func (c *Client) GetMediaInfo(url string) (*domain.MediaInfo, error) {
+	if cached := c.getCached(url); cached != nil {
+		slog.Debug("cache hit", "url", url)
+		return cached, nil
+	}
+
 	args := []string{
 		"-J", "--no-download",
 		"--no-warnings",
@@ -106,6 +146,7 @@ func (c *Client) GetMediaInfo(url string) (*domain.MediaInfo, error) {
 	if mi.IsPlaylist {
 		mi.PlaylistCount = info.PlaylistCount
 		mi.PlaylistEntries = c.parsePlaylistEntries(info.Entries)
+		c.setCache(url, mi)
 		return mi, nil
 	}
 
@@ -124,11 +165,17 @@ func (c *Client) GetMediaInfo(url string) (*domain.MediaInfo, error) {
 	}
 	mi.EstimatedSize = maxSize
 
+	c.setCache(url, mi)
 	return mi, nil
 }
 
 // GetPlaylistInfo fetches a flat playlist listing (fast, no format detail).
 func (c *Client) GetPlaylistInfo(url string) (*domain.MediaInfo, error) {
+	if cached := c.getCached(url); cached != nil {
+		slog.Debug("cache hit", "url", url)
+		return cached, nil
+	}
+
 	args := []string{
 		"-J", "--flat-playlist", "--no-download",
 		"--no-warnings",
@@ -159,6 +206,7 @@ func (c *Client) GetPlaylistInfo(url string) (*domain.MediaInfo, error) {
 		PlaylistEntries: c.parsePlaylistEntries(info.Entries),
 	}
 
+	c.setCache(url, mi)
 	return mi, nil
 }
 
@@ -376,6 +424,36 @@ func (c *Client) SearchYoutube(query string) (url, title string, err error) {
 	return info.Entries[0].URL, info.Entries[0].Title, nil
 }
 
+// Search does a yt-dlp search and returns entries.
+func (c *Client) Search(query, source string, limit int) ([]domain.PlaylistEntry, error) {
+	searchPrefix := "ytsearch"
+	if source == "ytmusic" {
+		searchPrefix = "ytmusicsearch"
+	}
+
+	args := []string{
+		"--flat-playlist", "-J", "--no-download",
+		"--no-warnings",
+		fmt.Sprintf("%s%d:%s", searchPrefix, limit, query),
+	}
+	if c.cookiesFile != "" {
+		args = append(args, "--cookies", c.cookiesFile)
+	}
+
+	slog.Info("searching", "query", query, "source", source)
+	out, err := c.run(args)
+	if err != nil {
+		return nil, fmt.Errorf("search failed: %w", err)
+	}
+
+	var info ytdlpInfo
+	if err := json.Unmarshal(out, &info); err != nil {
+		return nil, fmt.Errorf("parse search result: %w", err)
+	}
+
+	return c.parsePlaylistEntries(info.Entries), nil
+}
+
 func (c *Client) run(args []string) ([]byte, error) {
 	cmd := exec.Command(c.binaryPath, args...)
 	var stderr bytes.Buffer