
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 numbers \(a_1, a_2, a_3, \ldots, a_n, a_{n+1}\) are terms of a geometric sequence, it must be shown that the expressions \(a_1^2 - a_2^2, a_2^2 - a_3^2, a_3^2 - a_4^2, \ldots\) also form a geometric sequence with their reciprocals. Calculate the sum of this sequence.

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