
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 an infinite sequence of positive real numbers $\left\{x_{n}\right\}$ satisfying $x_{0}=1$ and $x_{i} \geq x_{i+1}$ for all $i \in \mathbb{N}$, such a sequence is called a "good sequence." Find the smallest constant $c$ such that there exists a "good sequence" $\left\{x_{n}\right\}$ for which the inequality $\sum_{i=0}^{n} \frac{x_{i}^{2}}{x_{i+1}} \leq c$ holds for any positive integer $n$.

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