
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 the operation defined as \\( \begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc \\), find the complex number \\(z\\) that satisfies the condition \\( \begin{vmatrix} z & 1+2i \\ 1-i & 1+i \end{vmatrix} = 0 \\).

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