
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
Define the operation $\begin{vmatrix} a & b \\ c & d\end{vmatrix} =ad-bc$. For a complex number $z=x+yi$ ($x,y\in\mathbb{R}$) that satisfies $\begin{vmatrix} z & 1 \\ z & 2i\end{vmatrix} =3+2i$, the complex number $z$ equals \_\_\_\_\_\_.

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