
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 conjugate of complex number $z$ be $\bar{z}$. Given that $(1+2i)\bar{z}=4+3i$,  
$(1)$ find the complex number $z$ and $\dfrac{z}{\bar{z}}$; $(2)$ find the equation of the trajectory of the point corresponding to the complex number $z_1$ that satisfies $|z_1-1|=|z|$.

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