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```python
# AI Dev Tool: FastAPI Pydantic v2 Micro-SaaS Boilerplate with PostgreSQL Vector Search Extension
# =================================================================================================
# This is a production-ready starter boilerplate for building micro-SaaS applications using FastAPI, 
# Pydantic v2, and PostgreSQL with 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 the FastAPI application
app = FastAPI()

# Define the PostgreSQL database connection URL
SQLALCHEMY_DATABASE_URL = "postgresql://user:password@host:port/dbname"

# Create a database engine
engine = create_engine(SQLALCHEMY_DATABASE_URL)

# Create a configured "Session" class
SessionLocal = sessionmaker(autocommit=False, autoflush=False, bind=engine)

# Create a base class for declarative class definitions
Base = declarative_base()

# Define the User model
class User(Base):
    """User model"""

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