
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
a) On a digital clock, the hours are displayed using four digits. For example, when it shows 00:00, we know it is midnight, and when it shows 23:59, we know it is one minute to midnight. How many times a day do all four displayed digits show even numbers?

b) A clock, with hour, minute, and second hands, makes a "plim" sound every time one hand passes another on the face of the clock. The three hands only overlap exactly at noon or midnight. What is the total number of "plim" sounds registered in one day, in the period between 12:00:01 PM and 11:59:59 PM?

Present the answer in LaTex format: \boxed{Your answer}
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