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```python
# AI Dev Tool: FastAPI Pydantic v2 Micro-SaaS Boilerplate with PostgreSQL Vector Search Extension
# Description: A production-ready starter boilerplate for building micro-SaaS applications using FastAPI, Pydantic v2, and PostgreSQL Vector Search Extension.

from fastapi import FastAPI, Depends
from fastapi.responses import JSONResponse
from pydantic import BaseModel
from typing import List, Optional
from sqlalchemy import create_engine, Column, Integer, String
from sqlalchemy.ext.declarative import declarative_base
from sqlalchemy.orm import sessionmaker, Session
from sqlalchemy.exc import SQLAlchemyError
from psycopg2.extras import Json

# Initialize FastAPI app
app = FastAPI()

# Initialize PostgreSQL database connection
SQLALCHEMY_DATABASE_URL = "postgresql://user:password@host:port/dbname"
engine = create_engine(SQLALCHEMY_DATABASE_URL)
SessionLocal = sessionmaker(autocommit=False, autoflush=False, bind=engine)

Base = declarative_base()

# Define PostgreSQL Vector Search Extension model
class VectorSearchModel(Base):
    __tablename__ = "vector_search"
    id = Column(Integer, primary_key=True)
    vector = Column(Json)

# Create PostgreSQL Vector Search Extension table
Base.metadata.create_all(bind

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