
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
Let $\left\{a_{n}\right\}$ and $\left\{b_{n}\right\}$ be sequences of positive real numbers such that for any integer $n \geqslant 101$, the following holds:
$$
a_{n}=\sqrt{\frac{1}{100} \sum_{j=1}^{100} b_{n-j}^{2}}, \quad b_{n}=\sqrt{\frac{1}{100} \sum_{j=1}^{100} a_{n-j}^{2}}.
$$

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