
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 $\{a_n\}$ whose sum of the first $n$ terms is $S_n$, and $S_n + 1 = 2a_n$, find the maximum value of $n$ for which the inequality $a_1^2 + a_2^2 + \ldots + a_n^2 < 5 \times 2^{n+1}$ holds.

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