
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
Find all functions  $f: \mathbb{Q} \mapsto \mathbb{C}$  satisfying

(i) For any  $x_1, x_2, \ldots, x_{1988} \in \mathbb{Q}$ ,  $f(x_{1} + x_{2} + \ldots + x_{1988}) = f(x_1)f(x_2) \ldots f(x_{1988})$ .

(ii)  $\overline{f(1988)}f(x) = f(1988)\overline{f(x)}$  for all  $x \in \mathbb{Q}$ .

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