
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
Insert a cube with edge length $a$ into the space formed by four solid spheres, each with a radius of 1 and touching each other in pairs, in such a way that the cube can freely rotate in any direction. Determine the maximum value of $a$.

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