
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
In triangle \( \triangle ABC \) inscribed in a unit circle \( O \), the corresponding complex numbers of points \( A \), \( B \), and \( C \) are \( t_1 \), \( t_2 \), and \( t_3 \) respectively. Let \( P \) be any point on the circumcircle, corresponding to the complex number \( t \). The equation of the Simson line of point \( P \) with respect to \( \triangle ABC \) is:
$$
t z - s_3 \bar{z} = \frac{1}{2 t} \left( t^3 + s_1 t^2 - s_2 t - s_3 \right),
$$
where \( s_1 = t_1 + t_2 + t_3 \), \( s_2 = t_2 t_3 + t_3 t_1 + t_1 t_2 \), and \( s_3 = t_1 t_2 t_3 \).

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