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```python
# AI Dev Tool: FastAPI PostgreSQL Vector Search Extension Boilerplate
# =====================================================================

from fastapi import FastAPI, HTTPException
from fastapi.responses import JSONResponse
from fastapi.requests import Request
from pydantic import BaseModel
from typing import List, Optional
import httpx
import psycopg2
from psycopg2 import Error
import logging

# Initialize the FastAPI application
app = FastAPI()

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

# Define the PostgreSQL vector search extension model
class VectorSearchModel(BaseModel):
    id: int
    vector: List[float]

# Define the PostgreSQL vector search extension repository
class VectorSearchRepository:
    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:
            self.conn = psycopg2.connect(
                host=self.host,
                database=self.db,
                user=self.user,
                password=self.password

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