
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 that $O$ is the coordinate origin, the complex numbers $z_1$ and $z_2$ correspond to the vectors $\overrightarrow{OZ_1}$ and $\overrightarrow{OZ_2}$, respectively. $\bar{z_1}$ is the complex conjugate of $z_1$. The vectors are represented as $\overrightarrow{OZ_1} = (10 - a^2, \frac{1}{a + 5})$ and $\overrightarrow{OZ_2} = (2a - 5, 2 - a)$, where $a \in \mathbb{R}$, and $(z_2 - z_1)$ is a purely imaginary number.

(1) Determine the quadrant in which the point corresponding to the complex number $\bar{z_1}$ lies in the complex plane.
(2) Calculate $|z_1 \cdot z_2|$.

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