
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 ${z}_{1}={m}^{2}+\frac{1}{m+1}i$ and ${z}_{2}=(2m-3)+\frac{1}{2}i$, where $m\in R$ and $i$ is the imaginary unit, and ${z}_{1}+{z}_{2}$ is a purely imaginary number.

(I) Find the value of the real number $m$.
(II) Find the value of ${z}_{1}\cdot {\overline{z}}_{2}$.

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