
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 \(z_1, z_2, z_3, ..., z_{12}\) be the 12 zeros of the polynomial \(z^{12} - 3^{12}\).
For each \(j\), let \(w_j\) be one of \(\frac{1}{2}z_j\), \(z_j\), or \(2z_j\).
Find the maximum possible value of the real part of
\[
\sum_{j=1}^{12} w_j.
\]

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