feat(tiles): right-click Tiles is a 2 / 4 / 6 count menu

Owner decision 10 ("give me then the option to choose only HOW many
tiles, 2,4,6 default is 6 so the menu is easier"; asked where it lives:
"Click opens 6"). The click still opens the grid at once; right-click,
Shift+F10 or the Menu key on the button opens a small menu of three
counts, each drawn as the grid's own layout (2x1, 2x2, 3x2), the
remembered one checked. It replaces the session picker, which is gone
(method, markup hook, CSS, zh-CN entries).

- A pick is remembered per device in codeman:tile-count (default 6;
  codeman:tile-grid stays ids only) and opens that many tiles. The click
  and Ctrl+Shift+G then open with it, at most what the window fits.
- Which sessions: the rule the click already had (the grid this tab
  last had, else an open split's two, else tab order, the active one
  included and focused), trimmed from the end (the focused one kept) or
  filled from tab order to the count. A remembered grid comes back with
  its tiles first, in their cells, holes filled first, then tab order
  (owner answer, superseding decision 8's "exactly the stored set"). A
  page-load restore still brings back exactly what was stored.
- With the grid open a pick re-forms it: a count change is a shape
  change under the cell model's rule (reformTileCells), the focused tile
  always kept, every joining tile mounted and laid out before any of
  them connects, so each fits once and sends one PTY resize.
- Ctrl/Cmd+click on a tab with the grid closed opens the count in total,
  that session among them and focused (owner answer: N, not N+1).
- A count the window cannot fit is greyed out with the reason; a
  remembered one stays checked, the keyboard starts on the largest that
  fits. Arrows, Home/End, Enter or Space; Escape closes the menu alone
  (the global handler gives it the key first, like the tab-group menu)
  and puts the keyboard back on the Tiles button; Tab and a click
  elsewhere close it.
- zh-CN for every new string (N 个窗格, 窗格数量, the titles, the Help
  modal row); the i18n test harvests the menu now, with a session named
  "6 tiles" as the user-text trap.

Tests: tile-grid-picker.test.ts becomes tile-grid-count-menu.test.ts
(the button checks kept, every picker check carried over to the menu,
plus keyboard, remembered count, re-form, Ctrl/Cmd+click and the batch
connect); the open-set, cap, restore, shortcuts, split-coexistence and
i18n expectations follow the count; the Help modal test escapes its
label (the new one has parentheses). The vm harness tracks
document.activeElement and makes SVG elements.

Co-Authored-By: Claude Opus 5.5 (1M context) <noreply@anthropic.com>

src/web/public/styles.css

@@ -19766,10 +19766,8 @@ body.tile-grid-resizing--row * {
   text-align: center;
 }
 
-/* The grid's accent buttons: the Attach overlay's, and the picker's Open
-   (which narrows the padding in its own block below). */
-.tile-attach-btn,
-.tile-picker-open {
+/* The Attach overlay's accent button. */
+.tile-attach-btn {
   padding: 4px 14px;
   font: inherit;
   color: var(--accent-ink, #fff);
@@ -19779,8 +19777,7 @@ body.tile-grid-resizing--row * {
   cursor: pointer;
 }
 
-.tile-attach-btn:disabled,
-.tile-picker-open:disabled {
+.tile-attach-btn:disabled {
   opacity: 0.5;
   cursor: default;
 }
@@ -19802,70 +19799,95 @@ body.tile-grid-resizing--row * {
   pointer-events: none;
 }
 
-/* Tiles picker (tile-grid.js openTilePicker): a checkbox per open session,
-   positioned `fixed` under the Tiles button; same layer as the split picker. */
-.tile-picker-menu {
+/* The Tiles button's right-click menu (tile-grid.js openTileCountMenu): how
+   many tiles, 2, 4 or 6, each with its shape; positioned `fixed` under the
+   button, on the split picker's layer and in its floating style. */
+.tile-count-menu {
   position: fixed;
   z-index: 1000;
-  min-width: 240px;
-  max-width: 340px;
-  max-height: 60vh;
+  min-width: 168px;
   display: flex;
   flex-direction: column;
+  gap: 2px;
   padding: 6px;
-  background: var(--bg-secondary, #1a1a1a);
-  border: 1px solid var(--border-color, #333);
-  border-radius: 6px;
-  box-shadow: 0 8px 24px rgba(0, 0, 0, 0.35);
+  background: var(--floating-bg, var(--bg-secondary, #1a1a1a));
+  backdrop-filter: blur(20px);
+  -webkit-backdrop-filter: blur(20px);
+  border: 1px solid var(--control-border, var(--border-color, #333));
+  border-radius: 10px;
+  box-shadow: 0 8px 32px rgba(0, 0, 0, 0.45), 0 2px 8px rgba(0, 0, 0, 0.25);
   font-size: 13px;
 }
 
-.tile-picker-list {
-  overflow-y: auto;
-  min-height: 0;
+.tile-count-heading {
+  padding: 2px 8px 4px;
+  font-size: 11px;
+  letter-spacing: 0.04em;
+  text-transform: uppercase;
+  color: var(--text-muted, #888);
 }
 
-.tile-picker-item {
+.tile-count-item {
   display: flex;
   align-items: center;
-  gap: 8px;
-  padding: 4px 6px;
-  border-radius: 4px;
+  gap: 10px;
+  width: 100%;
+  padding: 6px 8px;
+  border: none;
+  border-radius: 6px;
+  background: none;
+  color: inherit;
+  font: inherit;
+  text-align: left;
   cursor: pointer;
 }
 
-.tile-picker-item:hover {
-  background: color-mix(in srgb, var(--text-muted, #888) 14%, transparent);
+.tile-count-item:hover:not(:disabled),
+.tile-count-item:focus {
+  outline: none;
+  background: color-mix(in srgb, var(--text-muted, #888) 16%, transparent);
 }
 
-.tile-picker-item span {
-  overflow: hidden;
+.tile-count-item:disabled {
+  opacity: 0.4;
+  cursor: default;
+}
+
+/* The shape: the grid's own layout for that count, as outlined cells; the
+   checked count's cells are filled with the accent. */
+.tile-count-glyph {
+  flex: 0 0 auto;
+  width: 30px;
+  height: 20px;
+  fill: none;
+  stroke: currentColor;
+  stroke-width: 1.2;
+  opacity: 0.75;
+}
+
+.tile-count-item[aria-checked='true'] .tile-count-glyph {
+  fill: color-mix(in srgb, var(--accent, #4a9eff) 35%, transparent);
+  stroke: var(--accent, #4a9eff);
+  opacity: 1;
+}
+
+.tile-count-label {
+  flex: 1 1 auto;
   white-space: nowrap;
-  text-overflow: ellipsis;
 }
 
-.tile-picker-footer {
-  display: flex;
-  align-items: center;
-  justify-content: space-between;
-  gap: 8px;
-  padding: 6px 4px 2px;
-  border-top: 1px solid var(--border-color, #333);
-  margin-top: 4px;
+.tile-count-check {
+  flex: 0 0 1em;
+  color: var(--accent, #4a9eff);
+  text-align: center;
 }
 
-.tile-picker-hint,
-.tile-picker-empty {
-  color: var(--text-muted);
-  font-size: 12px;
-}
-
-.tile-picker-empty {
-  padding: 6px 8px;
-}
-
-.tile-picker-open {
-  padding: 4px 12px;
+.tile-count-hint {
+  padding: 4px 8px 2px;
+  border-top: 1px solid var(--control-border, var(--border-color, #333));
+  margin-top: 2px;
+  color: var(--text-muted, #888);
+  font-size: 11px;
 }
 
 /* Split-picker menu: a small popover listing sessions to split with, appended