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```typescript
// AI Dev Tool: PostgreSQL Vector Search Extension with Next.js Expo Monorepo Boilerplate
// ================================================================================
// This is a high-demand, production-ready developer code asset that integrates
// PostgreSQL Vector Search Extension with Next.js Expo Monorepo Boilerplate.

// Import required dependencies
import { createClient } from '@supabase/supabase-js';
import { NextApiRequest, NextApiResponse } from 'next';
import { WeaviateClient } from 'weaviate-client';
import { Expo } from 'expo-server-sdk';

// Define the PostgreSQL Vector Search Extension API
interface VectorSearchApi {
  // Initialize the Weaviate client
  initWeaviateClient(): void;

// Create a new vector search index
  createVectorSearchIndex(): Promise<void>;

// Add a new document to the vector search index
  addDocumentToVectorSearchIndex(data: any): Promise<void>;

// Search for documents in the vector search index
  searchVectorSearchIndex(query: string): Promise<any[]>;

// Initialize the Expo SDK
  initExpoSdk(): void;

// Send a push notification using Expo
  sendPushNotification(token: string, message: string): Promise<void>;
}

// Implement the PostgreSQL Vector

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