
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
Given a sequence of complex numbers \( z_n \) defined such that for every positive integer \( n \), the following conditions hold:
\[ z_n \cdot \overline{z_{n+1}} = z_{n^2 + n + 1} \]
\[ z_n \cdot z_{n+2} = z_{n+1}^2 - 1 \]
Given \( z_2 = 2 + i \), find the value of \( z_9 \).

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