Min Cost Climbing Stairs

Find the minimum paid cost to step past the top when each move climbs one or two stairs.

A good solution should

• State the invariant or decision rule that makes the approach correct.
• Explain time and space costs in terms of input size and required output.
• Handle empty, minimal, duplicate, and boundary-shaped inputs.

Reasoning prompt

Define the allowed starting positions, then update the cheaper cost to reach each later position.

Use the linked judge for the canonical challenge. The examples above are compact TRT checks; create additional tests around the boundary most likely to break your invariant.