#34513 [Feature]: Add `litellm doctor` CLI subcommand for SDK diagnostic checks
## Problem
Every litellm support thread starts the same way: "is your API key set?", "is your model name correct?", "what version of litellm are you on?", "is your `.env` being loaded?". Today the answer requires manually checking Python imports, env vars, and `model_prices_and_context_window.json` for a user who often doesn't know where to look. A single `litellm doctor` command that runs a fixed battery of checks and prints the results would cut a lot of this back-and-forth and would also catch misconfigurations the user didn't know to look for (a stale `model_prices_and_context_window.json`, a missing optional dep that a particular provider needs, etc.).
## Motivation
A `doctor` subcommand is a low-risk, high-utility developer tool. It does not change runtime behavior; it only reads configuration and reports. The litellm CLI already exists at `litellm/proxy/client/cli/` (`lite`, `litellm-proxy` entry points) but those are for **managing a proxy server**. There is no diagnostic command today, no issue tracking it, and no equivalent helper module.
## Background
- The existing CLI is in `litellm/proxy/client/cli/` (click + rich). It is for proxy administrators, not for SDK use…