
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 that the complex number $z\_1$ satisfies $z\_1(2+i)=5i$ ($i$ is the imaginary unit), if the complex number $z\_2$ satisfies that $z\_1+z\_2$ is a real number and $z\_1⋅z\_2$ is a purely imaginary number, find the complex number $z\_2$.

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