
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
1. Given that $z_1, z_2 \in \mathbb{C}$, if $|z_1| = 5$ and $z_2 = 3 + 4i$, and $z_1 \cdot \overline{z_2}$ is a pure imaginary number, find $z_1$.
2. In the parallelogram ABCD, points A, B, and C correspond to the complex numbers $2+i$, $4+3i$, and $3+5i$, respectively. Find the complex number corresponding to point D.

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