
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
Given that the edge length of the cube $A_1A_2A_3A_4-B_1B_2B_3B_4$ is 1, find the number of elements in the set $\{x|x= \overrightarrow{A_{1}B_{1}} \cdot \overrightarrow{A_{i}B_{j}}, i \in \{1,2,3,4\}, j \in \{1,2,3,4\}\}$.

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