
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]


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Given a sequence of positive numbers \(\{a_n\}\) where \(a_1=1\), \(a_2=2\), and \(2a_n^2=a_{n-1}^2+a_{n+1}^2 (n\geqslant 2)\), let \(b_n=\frac{1}{a_n+a_{n+1}}\). Denote the sum of the first n terms of the sequence \(\{b_n\}\) as \(S_n\), calculate the value of \(S_{320}\).

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