
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
The function
\[f(z) = \frac{(2 + i \sqrt{2}) z + (\sqrt{2} + 4i)}{2} + 1 - 2i\]
represents a transformation in the complex plane consisting of a rotation and translation. Find the fixed point $c$ where $f(c) = c$.

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