
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 $z_1=a+3i$ ($a\in \mathbb{R}$), $z_2=3-4i$, and the product of $z_1$ and $z_2$ is a pure imaginary number.

(1) Find the value of the real number $a$; (2) If $z=1+\frac{z_1-z_2}{z_1+z_2}$, find $\bar{z}$ and $|z|$.

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