
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 complex numbers \( z \) and \( w \) satisfy \( |z| = 3 \) and \( (z + \bar{w})(\bar{z} - w) = 7 + 4i \), where \( i \) is the imaginary unit, and \( \bar{z} \) and \( \bar{w} \) denote the conjugates of \( z \) and \( w \), respectively. Find the modulus of \( (z + 2\bar{w})(\bar{z} - 2w) \).

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