
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
For any real sequence $\left\{x_{n}\right\}$, define the sequence $\left\{y_{n}\right\}$ as follows:
$$
y_{1} = x_{1}, \quad y_{n+1} = x_{n+1} - \left(\sum_{i=1}^{n} x_{i}^{2}\right)^{\frac{1}{2}} \quad (n \geqslant 1).
$$

Find the smallest positive number $\lambda$ such that for any real sequence $\left\{x_{n}\right\}$ and any positive integer $m$, the following inequality holds:
$$
\frac{1}{m} \sum_{i=1}^{m} x_{i}^{2} \leqslant \sum_{i=1}^{m} \lambda^{m-i} y_{i}^{2}.
$$

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