When the options depend on a mode

The second dimension is not a position. It is which situation you are currently in.

Why it works

Design the modes first, on paper

List the situations you can be in, and for each, list what you may do and where it takes you. Do that before writing anything, because the code is a direct transcription of that list and is very hard to derive without it.

Three modes cover a process where you may hold something, have just released it, or be idle: holding, just released, and free to act. Just released is a mode rather than a moment because it restricts what may happen next.

Write the list out as data before writing any loop. Doing it this way makes the missing arrow obvious: nothing goes from just released straight back to holding, which is exactly what the pause means.

1# For each mode: what you may do this step, and the mode it leaves you in.2transitions = {3    "holding":     [("keep holding", "holding"), ("release", "just released")],4    "just released": [("must idle", "free")],5    "free":        [("stay idle", "free"), ("take one", "holding")],6}7 8for mode, options in transitions.items():9    for action, lands_in in options:10        print(f"{mode:15} {action:14} -> {lands_in}")

Five arrows, so five terms in the update. Each one becomes a max or an addition.

Quick check· 5 min

Why is just-released its own mode?

In a process where the step after releasing must be idle, why model just-released separately rather than merging it with free?

Because the choices available differ: from just-released you may not act, while from free you mayBecause it makes the table rectangular, which simplifies the loopBecause merging them would double the time complexityIt does not need to be separate; two modes are sufficient

Problem· 30 min

Best Time To Buy And Sell Stock With Cooldown

Read the constraints first and let them tell you what complexity is expected. Derive the approach, implement it, run the tests, and submit when it passes.

1# For each mode: what you may do this step, and the mode it leaves you in.2transitions = {3 "holding": [("keep holding", "holding"), ("release", "just released")],4 "just released": [("must idle", "free")],5 "free": [("stay idle", "free"), ("take one", "holding")],6}7 8for mode, options in transitions.items():9 for action, lands_in in options:10 print(f"{mode:15} {action:14} -> {lands_in}")