
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
If the sequence $\{a_{n}\}$ satisfies ${a}_{n}^{2}-{a}_{n-1}^{2}=p$ ($n\geqslant 2$, $n\in N^{*}$, $p$ is a constant), then the sequence $\{a_{n}\}$ is called an "equi-variance sequence." Let $S_{n}$ be the sum of the first $n$ terms of the positive sequence $\{a_{n}\}$. It is known that $\{S_{n}\}$ is an "equi-variance sequence," and ${S}_{2}+{S}_{4}=2+\sqrt{2}$, ${a}_{3}+{a}_{4}=2-\sqrt{2}$. Find the minimum value of $a_{n}+S_{n}$.

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