
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 sequence $\{a_{n}a_{n+1}\}$ is a geometric sequence with a common ratio of $2$, $a_{1}=1$, $a_{2}=2$. Let $S_{n}$ denote the sum of the first $n$ terms of the sequence $\{a_{n}\}$. If the inequality $\frac{{{S_{2n+1}}+2}}{{4{a_{2n}}}}>\frac{{x-1}}{x}$ holds for any $x\in \left(0,2023\right]$, then the minimum value of $n$ is ____.

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