
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
The robot vacuum cleaner is programmed to move on the floor according to the law:

$$\left\{\begin{array}{l} x = t(t-6)^2 \\ y = 0, \quad 0 \leq t \leq 7 \\ y = (t-7)^2, \quad t \geq 7 \end{array}\right.$$

where the axes are chosen parallel to the walls and the movement starts from the origin. Time $t$ is measured in minutes, and coordinates are measured in meters. Find the distance traveled by the robot in the first 7 minutes and the absolute change in the velocity vector during the eighth minute.

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