
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
Let the complex number $z_{1}=2+ai$ (where $a\in\mathbb{R}$), $z_{2}=3-4i$.

$(1)$ If $a=1$, find the value of $z_{1}z_{2}$.

$(2)$ If $z_{1}+z_{2}$ is a real number, find the value of $a$.

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