⚡ Live Sandbox: AI Dev Tool: PostgreSQL Vector Search Extension

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```python
# AI Dev Tool: FastAPI Pydantic v2 Micro-SaaS Starter
# Description: A production-ready starter boilerplate for building micro-SaaS applications with 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 Optional
from psycopg2 import connect
from psycopg2.extras import DictCursor
from cloudflare import WorkersKV

# Define the application
app = FastAPI()

# Define the database connection settings
DB_HOST = "localhost"
DB_PORT = 5432
DB_USER = "postgres"
DB_PASSWORD = "postgres"
DB_NAME = "micro_saaS"

# Define the PostgreSQL Vector Search Extension connection
def get_db_connection():
    """Get a connection to the PostgreSQL database."""
    return connect(
        host=DB_HOST,
        port=DB_PORT,
        user=DB_USER,
        password=DB_PASSWORD,
        dbname=DB_NAME,
        cursor_factory=DictCursor
    )

# Define the Workers KV store
workers_kv = WorkersKV(namespace="micro_saaS")

# Define the user model
class User(BaseModel):
    """User model."""
    id: int
    name:

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