
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 the sequence $\{a_n\}$, with the property that $a_{1}+2a_{2}+2^{2}a_{3}+\ldots+2^{n-1}a_{n}=2^{2n}-1$ for any natural number $n$, calculate $a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+\ldots+a_{n}^{2}$.

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