
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
There exists complex numbers  $z=x+yi$  such that the point  $(x, y)$  lies on the ellipse with equation  $\frac{x^2}9+\frac{y^2}{16}=1$ . If  $\frac{z-1-i}{z-i}$  is real, compute  $z$ .

*(Source: China National High School Mathematics League 2021, Zhejiang Province, Problem 3)*

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