
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 \(a_{1}, a_{2}, \ldots, a_{n}, a_{n+1}\) be distinct elements of an arithmetic sequence. Write the sum 

\[ S_{n} = x^{a_{2}^{2}-a_{1}^{2}} + x^{a_{3}^{2}-a_{2}^{2}} + \ldots + x^{a_{n+1}^{2}-a_{n}^{2}} \]

in a simplified form, given that \(x \neq 1\).

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