
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
Let \( Z \) be a complex number, and \( A \), \( B \), \( C \) be real numbers with \( A < 0 \). Define \( f(Z) = A|Z|^2 + 2B \cdot \operatorname{Re} Z + C \) (where \(\operatorname{Re} Z\) denotes the real part of \( Z \)). Determine the value of \( Z \) for which \( f(Z) \) achieves its maximum value.

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