feat(server): add open-in-terminal endpoint

Add POST /open-in-terminal endpoint to open a system terminal in the
worktree directory using the cross-platform openInTerminal utility.

The endpoint validates that worktreePath is provided and is an
absolute path for security.

Extracted from PR #558.

apps/server/src/routes/worktree/index.ts

@@ -29,6 +29,7 @@ import {
   createGetAvailableEditorsHandler,
   createRefreshEditorsHandler,
 } from './routes/open-in-editor.js';
+import { createOpenInTerminalHandler } from './routes/open-in-terminal.js';
 import { createInitGitHandler } from './routes/init-git.js';
 import { createMigrateHandler } from './routes/migrate.js';
 import { createStartDevHandler } from './routes/start-dev.js';
@@ -97,6 +98,11 @@ export function createWorktreeRoutes(
   );
   router.post('/switch-branch', requireValidWorktree, createSwitchBranchHandler());
   router.post('/open-in-editor', validatePathParams('worktreePath'), createOpenInEditorHandler());
+  router.post(
+    '/open-in-terminal',
+    validatePathParams('worktreePath'),
+    createOpenInTerminalHandler()
+  );
   router.get('/default-editor', createGetDefaultEditorHandler());
   router.get('/available-editors', createGetAvailableEditorsHandler());
   router.post('/refresh-editors', createRefreshEditorsHandler());

apps/server/src/routes/worktree/routes/open-in-terminal.ts

@@ -0,0 +1,50 @@
+/**
+ * POST /open-in-terminal endpoint - Open a terminal in a worktree directory
+ *
+ * This module uses @automaker/platform for cross-platform terminal launching.
+ */
+
+import type { Request, Response } from 'express';
+import { isAbsolute } from 'path';
+import { openInTerminal } from '@automaker/platform';
+import { getErrorMessage, logError } from '../common.js';
+
+export function createOpenInTerminalHandler() {
+  return async (req: Request, res: Response): Promise<void> => {
+    try {
+      const { worktreePath } = req.body as {
+        worktreePath: string;
+      };
+
+      if (!worktreePath) {
+        res.status(400).json({
+          success: false,
+          error: 'worktreePath required',
+        });
+        return;
+      }
+
+      // Security: Validate that worktreePath is an absolute path
+      if (!isAbsolute(worktreePath)) {
+        res.status(400).json({
+          success: false,
+          error: 'worktreePath must be an absolute path',
+        });
+        return;
+      }
+
+      // Use the platform utility to open in terminal
+      const result = await openInTerminal(worktreePath);
+      res.json({
+        success: true,
+        result: {
+          message: `Opened terminal in ${worktreePath}`,
+          terminalName: result.terminalName,
+        },
+      });
+    } catch (error) {
+      logError(error, 'Open in terminal failed');
+      res.status(500).json({ success: false, error: getErrorMessage(error) });
+    }
+  };
+}