
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 complex numbers $z_{1}=1-ai$ and $z_{2}=2a+3i$ where $a\in \mathbb{R}$.  
$(1)$ If $z_{1}z_{2}$ is purely imaginary, find the value of $|z_{1}+z_{2}|$.  
$(2)$ If the complex number $\frac{z_{2}}{z_{1}}$ corresponds to a point on the line $y=5x$ in the complex plane, find the value of $a$.

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