
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 time is chosen randomly and uniformly in an 24-hour day. The probability that at that time, the (non-reflex) angle between the hour hand and minute hand on a clock is less than  $\frac{360}{11}$  degrees is  $\frac{m}{n}$  for coprime positive integers  $m$  and  $n$ . Find  $100m + n$ .

*Proposed by Yannick Yao*

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