
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 $z$ satisfy the equation $|2z+15|=\sqrt{3} |\bar{z} +10|$.

(1) Calculate the value of $|z|$;

(2) Is there a real number $a$ such that $\frac{z}{a} + \frac{a}{z}$ is a real number? If yes, find the value of $a$; if not, explain why.

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