
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 edge length of the cube $A_{1}A_{2}A_{3}A_{4}-B_{1}B_{2}B_{3}B_{4}$ is $1$, calculate 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