
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 let the complex numbers \( z_{1}=a+\mathrm{i} \), \( z_{2}=2a+2\mathrm{i} \), and \( z_{3}=3a+4\mathrm{i} \), where \( \mathrm{i} \) is the imaginary unit. If \( \left|z_1\right| \), \( \left|z_2\right| \), and \( \left|z_3\right| \) form a geometric sequence, find the value of \( a \).

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