
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 ${z}_{1}=5+10i,{z}_{2}=3-4i$.

(1) If ${z}_{1}$ and ${z}_{2}$ correspond to points $A$ and $B$ in the complex plane, find the complex number that corresponds to the directed segment $\overrightarrow{AB}$.

(2) If $\frac{1}{z}= \frac{1}{{z}_{1}}+ \frac{1}{{z}_{2}}$, find $z$.

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