
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 complex number \( z \) satisfies
$$
4 z^{-2011}-3 \mathrm{i} z^{-2010}-3 \mathrm{i} z^{-1}-4=0 \text {, }
$$
find the range of values of \( t=\overline{\left(\frac{3-4 \mathrm{i}}{z}\right)}+\overline{(3+4 \mathrm{i}) z} \).

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