
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 $$\begin{vmatrix} a & c \\ b & d\end{vmatrix} = ad-bc$$. The complex number $z$ satisfies $$\begin{vmatrix} z & i \\ 1 & i\end{vmatrix} = 1+i$$. Let $\overset{ .}{z}$ be the conjugate of $z$. Then, $\overset{ .}{z} =$ \_\_\_\_\_\_.

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