The Ember-Pouches of the Skaldrin Icefield

Contract posted by the Hearthward Company, for any party bold enough to cross the Skaldrin Icefield.

The crossing takes many days, and the only thing between a traveller and the killing cold is the ember-pouch sewn into each coat. Every day on the ice changes the pouch's heat. A day of good fuel and shelter raises it, and a day of open wind drains it. The Company wants a reckoning of how many days a traveller spent with the pouch gone completely cold.

The pouch's heat (its *warmth*) is a whole number with a floor and a ceiling:

• It never rises above cap. Any heat beyond cap is lost.
• It never falls below 0. Any loss beyond 0 is ignored.

Write frozenDays(start, cap, days) where:

• start is an integer, the warmth before the first day, with 0 <= start <= cap.
• cap is a non-negative integer, the most warmth the pouch can hold.
• days is a list of integers, one per day in order. Each is that day's change in warmth. It may be positive, negative or zero.

Process the days in order. For each day:

1. Add the day's change to the current warmth. 2. If the result is greater than cap, set it to cap. 3. If the result is less than 0, set it to 0.

The warmth after these steps carries into the next day.

Return an integer: the number of days after which the warmth is exactly 0. The starting warmth is not a day and is never counted. If days is empty, return 0.