
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 straight path is 2 km in length. Beatrice walks from the beginning of the path to the end of the path at a constant speed of 5 km/h. Hieu cycles from the beginning of the path to the end of the path at a constant speed of 15 km/h.
(a) Suppose that Hieu starts 10 minutes later than Beatrice. Determine the number of minutes that it takes Hieu to catch up to Beatrice on the path.
(b) Suppose that Beatrice starts at a time that is exactly \( b \) minutes after 9:00 a.m. and that Hieu starts at a time that is exactly \( h \) minutes after 9:00 a.m., where \( b \) and \( h \) are integers from 0 to 59, inclusive, that are each randomly and independently chosen.
   (i) Determine the probability that there is a time at which Beatrice and Hieu are at the same place on the path. (Beatrice and Hieu are at the same place at a given time if they start at the same time or finish at the same time, or are at the same place at some point between the beginning and end of the path at the same time.)
   (ii) One day, Beatrice uses a scooter and travels from the beginning of the path to the end of the path at a constant speed of \( x \) km/h, where \( x > 5 \) and \( x < 15 \). Hieu still cycles from the beginning of the path to the end of the path at a constant speed of 15 km/h. If the probability that there is a time at which Beatrice and Hieu are at the same place on the path is \( \frac{13}{200} \), determine the range of possible values of \( x \).

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