
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 ball rolls down a smooth trough, curved in the shape of a cycloid (diagram 212). Ignoring friction and air resistance, find: 

1. The dependence of the path traveled by the center of gravity of the ball on time.
2. The time it takes for the ball to roll from the beginning of the trough to its lowest point: 
   a) along the trough
   b) along a straight line.

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