
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 \( z_{1}, z_{2}, \cdots, z_{12} \) be the roots of the equation \( z^{12}=2^{36} \). Let \( w_{j} (j=1,2, \cdots, 12) \) be one of \( z_{j} \) or \( iz_{j} \). Find the maximum value of \( \operatorname{Re} \sum_{j=1}^{12} w_{j} \).

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