
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
In a simple graph, a sequence $P_{1}, P_{2}, \ldots, P_{k}$ is called a "hopping school" if for every $i$, the degree of the vertex $P_{i}$ is $i$. What is the maximum number of vertices that a hopping school can have in a graph with $n$ vertices?

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