Problem 2431. Power Times (of the day)

Many times throughout the day can represent mathematical equations. In this problem, we focus on times that represent powers. For example, 8:23 can be written as 8=2^3. Write a function that determines if the given time (restricted to three digits in 12-hour time, 1:00 to 9:59) is a power time. There are four types that are categorized here, and a given time can fit more than one category:

 - equation written forward, "=" doesn't coincide with ":" --> add 1 to output (e.g., 2:38)

 - equation written forward, "=" does coincide with ":" -- > add 100 to output (e.g., 8:23)

 - equation written backward, "=" doesn't coincide with ":" --> add 10 to output (e.g., 3:28)

 - equation written backward, "=" does coincide with ":" --> add 1000 to output (e.g., 9:23)

Examples of combination times include 4:22 (1100 since 4=2^2 and 2^2=4) and 1:31 (1001 since 1^3=1 and 1^3=1).

This problem is related to Problem 2432 and Problem 2433.