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```python
# AI Dev Tool: FastAPI PostgreSQL Vector Search Micro-SaaS
# Description: A production-ready FastAPI micro-SaaS template with PostgreSQL vector search extension and Pydantic v2 validation.

from fastapi import FastAPI, HTTPException
from fastapi.responses import JSONResponse
from pydantic import BaseModel
from typing import List, Optional
import psycopg2
import psycopg2.extras
import json

# Define the FastAPI application
app = FastAPI(
    title="FastAPI PostgreSQL Vector Search Micro-SaaS",
    description="A production-ready FastAPI micro-SaaS template with PostgreSQL vector search extension and Pydantic v2 validation.",
    version="1.0.0",
)

# Define the PostgreSQL connection settings
POSTGRES_HOST = "localhost"
POSTGRES_DB = "vector_search"
POSTGRES_USER = "postgres"
POSTGRES_PASSWORD = "postgres"

# Define the PostgreSQL vector search extension
class VectorSearchExtension:
    def __init__(self, host: str, db: str, user: str, password: str):
        self.host = host
        self.db = db
        self.user = user
        self.password = password
        self.conn = None

def connect(self):
        try:

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