⚡ Live Sandbox: AI Dev Tool: FastAPI Pydantic v2 (dg11)

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```python
# AI Dev Tool: FastAPI Pydantic v2 Micro-SaaS Boilerplate with PostgreSQL Vector Search Extension
# =================================================================================================
# This is a high-demand, production-ready developer code asset 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 psycopg2 import connect
from psycopg2.extras import DictCursor
from sqlalchemy import create_engine, Column, Integer, String
from sqlalchemy.ext.declarative import declarative_base
from sqlalchemy.orm import sessionmaker
from sqlalchemy.dialects.postgresql import TSVECTOR

# Initialize the FastAPI application
app = FastAPI()

# Define the PostgreSQL database connection settings
POSTGRES_HOST = "localhost"
POSTGRES_PORT = 5432
POSTGRES_USER = "postgres"
POSTGRES_PASSWORD = "postgres"
POSTGRES_DB = "micro_saaS"

# Create a PostgreSQL database engine
engine = create_engine(f"postgresql://{POSTGRES_USER}:{POSTGRES_PASSWORD}@{POSTGRES_HOST}:{POSTGRES_PORT}/{POSTGRES_DB}")

# Define the base class for SQLAlchemy models

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