
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 ants, Gosha and Lesha, crawl along the diagonals \(AC\) and \(B_1D_1\) of the base of cube \(ABCD A_1B_1C_1D_1\) with edge length \(a\). They started moving simultaneously from points \(A\) and \(B_1\), respectively, with constant speeds, where Lesha's speed was four times that of Gosha's, 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}
user