
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
The base of a unit edge parallelepiped is $A_{1} A_{2} A_{3} A_{4}$, and the top face is $B_{1} B_{2} B_{3} B_{4}$ such that the vertices $A_{i}$ and $B_{i}$ are connected by an edge. What are the limits within which the sum of squares $A_{1} B_{2}^{2} + A_{2} B_{3}^{2} + A_{3} B_{4}^{2} + A_{4} B_{1}^{2}$ can vary?

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