
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 a sequence ${a_n}$ with its first term as $1$, $S_n$ represents the sum of the first $n$ terms of the sequence, and $S_{n+1} = qS_n + 1$, where $q > 0$, $n \in \mathbb{N}^*.$ Suppose the eccentricity of the hyperbola $x^2 - \frac{y^2}{a_n^2} = 1$ is $e_n$, and $e_2 = 2$, then find the sum $e_1^2 + e_2^2 + \ldots + e_n^2 =$ ___.

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