
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 $m\in\mathbb{R}$, $i$ is the imaginary unit, and $z_{1}+z_{2}$ is a purely imaginary number.

(1) Find the value of the real number $m$.
(2) Find the value of $z_{1}\cdot\overline{z_{2}}$.

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