
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 bumblebee flew towards one apple tree with a speed of \( v_{1} \) m/min. Simultaneously, a bee flew towards another apple tree with a speed of \( v_{2} \) m/min. The bumblebee had to cover a distance of \( 2a \) meters, while the bee had to cover a distance of \( 2b \) meters. Suppose their flight paths are mutually perpendicular lines intersecting at a point that halves the path of both the bumblebee and the bee. Find a formula that expresses the distance \( y \) between the bumblebee and the bee as a function of their flight time \( x \). Determine the moment when the distance between the bumblebee and the bee in flight reaches its minimum.

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