
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
Bus A leaves the terminus every 20 minutes, travels a distance of 1 mile to a circular road of length 10 miles, goes clockwise around the road, and then returns along the same road to the terminus (a total distance of 12 miles). The journey takes 20 minutes, and the bus travels at a constant speed. Upon reaching the terminus, it immediately repeats the journey. Bus B follows the same procedure except that it leaves the terminus 10 minutes after Bus A and travels the opposite direction around the circular road. The time taken to pick up or set down passengers is negligible.

A man wants to catch a bus at a distance \(0 < x < 12\) miles from the terminus (along the route of Bus A). Let \(f(x)\) be the maximum time his journey can take (waiting time plus journey time to the terminus). Find \(f(2)\) and \(f(4)\), and determine the value of \(x\) for which \(f(x)\) is a maximum. Sketch \(f(x)\).

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