
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 $|z\_1|=5$, $z\_2=3+4i$, and $z\_1 \cdot \bar{z\_2}$ is a purely 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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