feat: expose taskBudget SDK option as task_budget action input

[rsd-darshan]

Summary

Closes #1329.

The SDK has supported taskBudget since v0.2.84 but there was no way to set it from the action layer — passing --task-budget via claude_args was silently ignored because the CLI doesn't recognise that flag.

• Added task_budget input to both action.yml and base-action/action.yml
• Added taskBudget?: string to ClaudeOptions in base-action/src/run-claude.ts
• Wired it through parseSdkOptions as sdkOptions.taskBudget = { total: parseInt(...) } (same pattern as maxTurns)
• Passed INPUT_TASK_BUDGET env var from both action entrypoints (base-action/src/index.ts and src/entrypoints/run.ts)

Test plan

• Set task_budget: 200000 in a workflow using this action and confirm the SDK receives taskBudget: { total: 200000 } (visible in step debug output with show_full_output: true)
• Omitting task_budget leaves the field undefined — no change in existing behaviour
• Passing a non-numeric value is handled the same way as maxTurns (parseInt produces NaN, SDK rejects it at runtime)

[rsd-darshan]

Hey @brianatdetections — this PR implements Option B from your request. The task_budget input is wired directly to sdkOptions.taskBudget = { total: parseInt(...) } so the SDK sends the task-budgets-2026-03-13 beta header automatically. Would appreciate a quick test against your setup to confirm it behaves as expected before this gets merged.

[brianatdetections]

Hey @brianatdetections — this PR implements Option B from your request. The task_budget input is wired directly to sdkOptions.taskBudget = { total: parseInt(...) } so the SDK sends the task-budgets-2026-03-13 beta header automatically. Would appreciate a quick test against your setup to confirm it behaves as expected before this gets merged.

Thanks! Will update and test later this week, looks good though.

[brianatdetections]

This is ready for maintainer review. I verified all CI checks pass locally against the PR head (0bda0e8):

Check	Result
bun run typecheck	✅ pass (tsc --noEmit)
bun run format:check	✅ pass (all files match Prettier style)
bun test	✅ 700 passed / 0 failed (39 files)

The change is minimal (16 lines, no deletions) and mirrors the existing maxTurns wiring end-to-end. The one thing worth calling out: it assigns taskBudget: { total: parseInt(...) } into the SDK Options type, and the pinned SDK (@anthropic-ai/claude-agent-sdk@0.3.173) declares exactly that shape — taskBudget?: { total: number } (sdk.d.ts:1587, marked @alpha) — so it typechecks cleanly.

Closes #1329.

@ashwin-ant — since this is an external PR, CI is held pending a maintainer "Approve and run workflows", and the approvals so far are from non-write accounts so they don't satisfy branch protection. Could you approve the workflow run and take a review pass when you get a chance? Happy to address anything.