
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  $z_1,z_2,z_3,\dots,z_{12}$  be the 12 zeroes of the polynomial  $z^{12}-2^{36}$ . For each  $j$ , let  $w_j$  be one of  $z_j$  or  $i z_j$ . Then the maximum possible value of the real part of  $\displaystyle\sum_{j=1}^{12} w_j$  can be written as  $m+\sqrt{n}$  where  $m$  and  $n$  are positive integers. Find  $m+n$ .

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