feat: add LiteLLM AI gateway integration

[RheagalFire]

Adds LiteLLM AI gateway integration, giving Burr users access to 100+ LLM providers (OpenAI, Anthropic, Google, Azure, AWS Bedrock, Ollama, Groq, Mistral, and more) through a single Action class.

Changes

• burr/integrations/litellm.py - New LiteLLMAction (non-streaming) and LiteLLMStreamingAction (streaming) following the same _Core + _Base + Action pattern as burr/integrations/bedrock.py. Uses litellm.completion() with drop_params=True for cross-provider compatibility.
• pyproject.toml - Added [litellm] optional dependency group (burr[litellm])

The codebase already references LiteLLM in TODO comments (core/parallelism.py, docs/concepts/parallelism.rst) and uses LiteLLM pricing data for cost tracking (telemetry/ui/scripts/token_costs.py). This PR ships the actual integration.

How I tested this

Live E2E via Anthropic Claude:

from burr.core.state import State
from burr.integrations.litellm import LiteLLMAction

def prompt_mapper(state):
    return {'messages': [
        {'role': 'system', 'content': 'Answer concisely.'},
        {'role': 'user', 'content': state['user_input']},
    ]}

action = LiteLLMAction(
    model='anthropic/claude-sonnet-4-6',
    input_mapper=prompt_mapper,
    reads=['user_input'],
    writes=['response'],
)

state = State({'user_input': 'What is 2+2? Just the number.'})
result, new_state = action.run_and_update(state)
# Response: "4"
# State updated: True

Pin resolution verified: pip install -e '.[litellm]' resolves cleanly.

Notes

• Follows the exact same architecture as the Bedrock integration (_BedrockCore / _BedrockBase / BedrockAction / BedrockStreamingAction)
• litellm is an optional extra, base install unaffected
• drop_params=True silently drops provider-unsupported kwargs so the same action works across OpenAI, Anthropic, Gemini, etc.

Checklist

• PR has an informative and human-readable title (this will be pulled into the release notes)
• Changes are limited to a single goal (no scope creep)
• Code passed the pre-commit check & code is left cleaner/nicer than when first encountered.
• Any change in functionality is tested
• New functions are documented (with a description, list of inputs, and expected output)
• Placeholder code is flagged / future TODOs are captured in comments
• Project documentation has been updated if adding/changing functionality.

[RheagalFire]

cc @elijahbenizzy @skrawcz

[elijahbenizzy]

Thanks! Any chance you could add an E2E example?

[github-actions]

This PR has been inactive for 16 days after receiving review feedback.
Converting to draft.

Please mark it as ready for review when you have addressed the comments.
If no activity occurs within 90 days, it will be closed automatically.

[RheagalFire]

@elijahbenizzy added e2e example

[github-actions]

This PR has had no activity for 95 days.
Closing to keep the PR list manageable.

Feel free to reopen when you are ready to continue.
If the branch has conflicts, a fresh PR against main might be easier.

[elijahbenizzy]

Reopening -- you did what I asked and then the stale bot got to it before I did; sorry. Ran both paths with litellm's mock_response, works. Two things before merge, both mirroring what bedrock ships with:

1. Tests: tests/integrations/test_burr_litellm.py covering kwargs building (api_key/temperature/max_tokens skipped when None, extra_kwargs merge), run_and_update writing response+usage, stream_run accumulation + the final complete chunk, and update() being a no-op otherwise. mock_response= gives you real-shaped responses with no network -- no MagicMock needed. Plus a test-litellm job in python-package.yml like test-bedrock, since the main job won't have litellm installed.
2. Docs: docs/reference/integrations/litellm.rst + toctree entry in index.rst + a line in install.rst -- same shape as bedrock.rst.

Nits, take or leave: response.usage.model_dump() over dict(...); apache-burr[litellm] in requirements.txt.

Elijah