feat: add 'processing' agent state for tool calling

[devin-ai-integration]

Summary

Adds a new "processing" state to AgentState to distinguish when the agent is executing function tools vs thinking (LLM inference). This allows applications to show appropriate UI/animations when the agent is executing tool calls, which typically take time and are silent.

Changes:

• Added "processing" to the AgentState Literal type in events.py
• Updated two state transitions in agent_activity.py to emit "processing" instead of "thinking" when tool execution begins:
  • Pipeline LLM path (line 2058): after speech playout when there are tool outputs
  • Realtime model path (line 2548): when first tool starts executing
• Updated test_tool_call test to verify the new state transition sequence (now expects 7 state events instead of 6)

State transition during tool calls:
speaking → processing (tool execution) → thinking (LLM inference for response) → speaking → listening

Closes #4460

Review & Testing Checklist for Human

• Verify state name choice: Confirm "processing" is the preferred name vs alternatives like "tool_calling" - this affects the public API
• Breaking change assessment: Adding a new state to the Literal type may affect consumers who exhaustively match on agent states - consider if this needs a migration note or changelog entry
• Verify completeness: Confirm there are no other code paths where tool execution should emit "processing" state (only two locations were changed)
• End-to-end test: Test with an agent that uses function tools to verify the state transitions work correctly in both pipeline and realtime model paths

Notes

• Link to Devin run: https://livekit.devinenterprise.com/sessions/61e3a23a3671411ebe4ef5488b7ef8f0
• Requested by: @theomonnom

[devin-ai-integration]

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