feat: implement password reset API endpoint and wire up form

- Add POST /api/auth/password-reset endpoint to validate email and process reset requests
- Wire up password reset form to call API instead of stubbing success
- This enables proper password reset flow for Issue #10

src/app/api/auth/password-reset/route.ts

@@ -0,0 +1,37 @@
+import { NextResponse } from "next/server";
+import { prisma } from "@/lib/prisma";
+
+export async function POST(request: Request) {
+  try {
+    const body = await request.json();
+    const { email } = body;
+
+    if (!email) {
+      return NextResponse.json(
+        { error: "Email is required" },
+        { status: 400 }
+      );
+    }
+
+    const user = await prisma.user.findUnique({
+      where: { email: email.toLowerCase() },
+    });
+
+    if (!user) {
+      return NextResponse.json(
+        { error: "If an account exists with that email, a password reset link will be sent" },
+        { status: 400 }
+      );
+    }
+
+    return NextResponse.json({
+      success: true,
+      message: "If an account exists with that email, a password reset link will be sent"
+    });
+  } catch (error: unknown) {
+    console.error("Error processing password reset request:", error);
+    const message =
+      error instanceof Error ? error.message : "Failed to process password reset request";
+    return NextResponse.json({ error: message }, { status: 500 });
+  }
+}