
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
Given complex numbers \( z_{1} = (2 - a) + (1 - b)i \), \( z_{2} = (3 + 2a) + (2 + 3b)i \), and \( z_{3} = (3 - a) + (3 - 2b)i \), where \( a, b \in \mathbb{R} \). Find the value of \( 3a + 4b \) when \( |z_{1}| + |z_{2}| + |z_{3}| \) is minimized.

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