
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
Along the diagonals $AC$ and $B_{1}D_{1}$ of the base of a cube $ABCDEFGH$ with edge length $a$, two ants, Gosha and Lesha, are crawling. They started moving simultaneously from points $A$ and $B_{1}$ respectively at constant speeds, with Lesha's speed being three times the speed of Gosha, and finished when Lesha reached point $D_{1}$. What was the minimum distance between Gosha and Lesha during their movement?

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