
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 that the sequence $\{a_n\}$ is a geometric sequence with a common ratio of $2$, and $a_3 - a_1 = 6$, find the value of $\frac{1}{{{a_1}^2}} + \frac{1}{{{a_2}^2}} + \dots + \frac{1}{{{a_n}^2}}$.

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