
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 of the bases \( AC \) and \( B_1D_1 \) of the cube \( ABCD A_1B_1C_1D_1 \) with side length \( a \), two ants, Gosha and Lesha, are crawling. They started simultaneously from points \( A \) and \( B_1 \) respectively, with constant speeds. Lesha's speed was twice the speed of Gosha, and they 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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