Expose Prometheus metrics on a separate port

Metrics are served on METRICS_PORT (default 2112, "off" to disable), never on
the public port. Besides the Go runtime and process metrics this adds:

- http_requests_total and http_request_duration_seconds per route pattern,
  from an Echo middleware.
- musicserver_songs_requested_total and musicserver_songs_played_total by
  pick mode (random, low, classic). The mode is now stored with each queued
  song (migration 000006, not released yet).
- Library size, sync duration, last successful sync, broken songs from the
  last full sync, and the database connection pool.

Refs #35, #36, #37

Co-Authored-By: Claude Opus 5.5 <noreply@anthropic.com>

internal/backend/music.go

@@ -12,6 +12,7 @@ import (
 
 	"music-server/internal/db/repository"
 	"music-server/internal/logging"
+	"music-server/internal/metrics"
 
 	"github.com/google/uuid"
 	"github.com/jackc/pgx/v5"
@@ -21,6 +22,24 @@ import (
 // maxPickAttempts bounds how many broken songs are skipped before giving up.
 const maxPickAttempts = 100
 
+// PickMode is how a random song was picked. It is stored with queued songs and used as
+// the mode label of the song metrics.
+type PickMode string
+
+const (
+	PickRandom  PickMode = "random"
+	PickLow     PickMode = "low"
+	PickClassic PickMode = "classic"
+)
+
+func init() {
+	// Start every mode at 0, so the series exist before the first song.
+	for _, mode := range []PickMode{PickRandom, PickLow, PickClassic} {
+		metrics.SongsRequested.WithLabelValues(string(mode))
+		metrics.SongsPlayed.WithLabelValues(string(mode))
+	}
+}
+
 type SongInfo struct {
 	SoundtrackUuid   uuid.UUID `json:"SoundtrackUuid"`
 	Soundtrack       string    `json:"Soundtrack"`
@@ -145,7 +164,11 @@ func (m *MusicService) markPlayed(ctx context.Context, songs []repository.ListQu
 	if err := m.q.AddSoundtrackPlayedByUUID(ctx, song.SoundtrackUuid); err != nil {
 		return err
 	}
-	return m.q.AddSongPlayedByUUID(ctx, song.SongUuid)
+	if err := m.q.AddSongPlayedByUUID(ctx, song.SongUuid); err != nil {
+		return err
+	}
+	metrics.SongsPlayed.WithLabelValues(song.Mode).Inc()
+	return nil
 }
 
 func (m *MusicService) loadQueue(ctx context.Context, queueID uuid.UUID) (repository.Queue, []repository.ListQueueSongsRow, error) {
@@ -167,7 +190,7 @@ func (m *MusicService) RandomSong(ctx context.Context, queueID uuid.UUID) (strin
 	for i, s := range soundtracks {
 		candidates[i] = weightedSoundtrack{soundtrack: s, weight: 1}
 	}
-	return m.pickAndRemember(ctx, queueID, candidates)
+	return m.pickAndRemember(ctx, queueID, candidates, PickRandom)
 }
 
 // RandomSongLowChance picks a random song, favouring soundtracks played less than average.
@@ -176,23 +199,30 @@ func (m *MusicService) RandomSongLowChance(ctx context.Context, queueID uuid.UUI
 	if err != nil {
 		return "", err
 	}
-	return m.pickAndRemember(ctx, queueID, lowChanceWeights(soundtracks))
+	return m.pickAndRemember(ctx, queueID, lowChanceWeights(soundtracks), PickLow)
 }
 
-func (m *MusicService) pickAndRemember(ctx context.Context, queueID uuid.UUID, candidates []weightedSoundtrack) (string, error) {
+func (m *MusicService) pickAndRemember(
+	ctx context.Context, queueID uuid.UUID, candidates []weightedSoundtrack, mode PickMode,
+) (string, error) {
 	song, err := m.pickSong(ctx, candidates)
 	if err != nil {
 		return "", err
 	}
-	return m.remember(ctx, queueID, song)
+	return m.remember(ctx, queueID, song, mode)
 }
 
-// remember stores the song as the queue's last fetched song, for AddLatestToQueue.
-func (m *MusicService) remember(ctx context.Context, queueID uuid.UUID, song repository.Song) (string, error) {
-	err := m.q.SetQueueLastFetched(ctx, repository.SetQueueLastFetchedParams{Uuid: queueID, LastFetchedSongUuid: song.Uuid})
+// remember stores the song and how it was picked as the queue's last fetched song,
+// for AddLatestToQueue.
+func (m *MusicService) remember(ctx context.Context, queueID uuid.UUID, song repository.Song, mode PickMode) (string, error) {
+	modeName := string(mode)
+	err := m.q.SetQueueLastFetched(ctx, repository.SetQueueLastFetchedParams{
+		Uuid: queueID, LastFetchedSongUuid: song.Uuid, LastFetchedMode: &modeName,
+	})
 	if err != nil {
 		return "", err
 	}
+	metrics.SongsRequested.WithLabelValues(modeName).Inc()
 	return song.Path, nil
 }
 
@@ -212,7 +242,7 @@ func (m *MusicService) RandomSongClassic(ctx context.Context, queueID uuid.UUID)
 			}
 			continue
 		}
-		return m.remember(ctx, queueID, row.Song)
+		return m.remember(ctx, queueID, row.Song, PickClassic)
 	}
 	return "", ErrNoSongs
 }