
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
The path index of a graph $G$ is the minimum number of paths needed to visit each vertex of $G$ exactly once. Given a connected graph $G$, what is the maximum possible value of its path index, knowing that the largest set of pairwise non-adjacent vertices in $G$ is $n>1$ (independence number)?

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