
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  $ P(z) \equal{} z^3 \plus{} az^2 \plus{} bz \plus{} c$ , where  $ a$ ,  $ b$ , and  $ c$  are real. There exists a complex number  $ w$  such that the three roots of  $ P(z)$  are  $ w \plus{} 3i$ ,  $ w \plus{} 9i$ , and  $ 2w \minus{} 4$ , where  $ i^2 \equal{} \minus{} 1$ . Find  $ |a \plus{} b \plus{} c|$ .

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