
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 sum of the first $n$ terms of a geometric sequence ${{a}_{n}}$ is ${S}_{n}={2}^{n}-a$, find the value of ${{a}_{1}}^{2}+{{a}_{2}}^{2}+⋯+{{a}_{n}}^{2}$.

A) ${\left({2}^{n}-1\right)}^{2}$
B) $\frac{1}{3}\left({2}^{n}-1\right)$
C) ${4}^{n}-1$
D) $\frac{1}{3}\left({4}^{n}-1\right)$

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