
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
Let \( A = \{1, -1, \mathrm{i}, -\mathrm{i} \} \) (where \( \mathrm{i} \) is the imaginary unit), and \( f(x) \) be a function whose domain and range are both \( A \). Given that for any \( x, y \in A \), \( f(x y) = f(x) f(y) \), how many such functions \( f(x) \) satisfy this condition?

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