
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 $a$ is a real number, $z\_1=2-i$, and $z\_2=a+i$ ($i$ is the imaginary unit),
(1) If $a=1$, indicate the quadrant in which the point corresponding to $z\_1+ \overline {z\_2}$ in the complex plane is located;
(2) If $z\_1 \cdot z\_2$ is a purely imaginary number, find the value of $a$.

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