
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 the complex numbers \(z\) and \(w\) satisfy \(|z| = 3\) and \((z + \bar{w})(\bar{z} - w) = 7 + 4i\), where \(i\) is the imaginary unit and \(\bar{z}\), \(\bar{w}\) 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}
user