
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
Two points $A$ and $B$ are moving along two skew lines $a$ and $b$, respectively. The first point moves with a constant velocity $\vec{m}$, and the second with a constant velocity $\vec{n}$. What line does point $P \in$ $\epsilon[A B]$ describe, satisfying the condition $\frac{\overrightarrow{P B}}{\overrightarrow{A B}}=k$? (For vectors in space, the notation $\frac{\stackrel{\rightharpoonup}{a}}{\vec{b}}$ has the same meaning as in § 2, problem 4.)

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