
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 $z_1$, $z_2$, $z_3$, $\dots$, $z_{10}$ be the 10 zeroes of the polynomial $z^{10} - 2^{50}$. For each $j$, let $w_j$ be one of $z_j$ 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}
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