
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 \( a \in \mathbf{R} \), and complex numbers 
\[ z_{1} = a + \mathrm{i}, \quad z_{2} = 2a + 2\mathrm{i}, \quad z_{3} = 3a + 4\mathrm{i}. \]
If \( \left|z_{1}\right|, \left|z_{2}\right|, \left|z_{3}\right| \) form a geometric sequence, find the value of the real number \( a \).

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