
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

There is a right angle. Point $A$ moves along one leg with speed $v$, and point $B$ moves along the other leg with speed $v_{1}$ towards the vertex. The distance of point $A$ from the vertex is $a$, and the distance of point $B$ from the vertex is $b$.

1. Determine when the distance between the two points is the smallest, and what this minimum distance is.
2. What is the distance of the points from the vertex at this moment?
3. What are the results if $v=v_{1}$?

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