
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
In the coordinate axes, the graph of the velocity versus time dependencies results in a trapezoid with bases \(V\) and \(U\). Since the distance traveled is the area under the velocity graph, the task is reduced to finding a segment parallel to the bases of the trapezoid that divides the area in the ratio \(1:2 (1:k)\). The desired velocity is given by

\[ 
W = \frac{V^2 + kU^2}{1+k} 
\]

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