
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 $\frac {1}{1-i}+ \frac {1}{2+3i}=x+yi$, find the values of the real numbers x and y.
2. Known that $z_1, z_2 \in \mathbb{C}$, if $z_1 =3+4i$, $|z_2|=5$, and $z_1 \cdot z_2$ is a purely imaginary number, find $z_2$.

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