
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 $z=\frac{1}{2}+\frac{\sqrt{3}}{2}i$, and $${(x-z)^{4}}={a_{0}}{x^{4}}+{a_{1}}{x^{3}}+{a_{2}}{x^{2}}+{a_{3}}x+{a_{4}}$$, find the complex number ${a_{2}}$.

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