
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 \( z_1, z_2, \ldots, z_{10} \) be the 10 zeroes of the polynomial \( z^{10} - 2^{30} \). Each zero \( z_j \) can be rotated by \( \frac{\pi}{2} \) (multiplied by \( i \)) to form \( w_j \), which is either \( z_j \) or \( iz_j \). Find the maximum possible value of the real part of
\[
\sum_{j=1}^{10} w_j.
\]

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