
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 \( G \) be a simple graph with \( k \) connected components, which have \( a_1, \ldots, a_k \) vertices, respectively. Determine the number of ways to add \( k-1 \) edges to \( G \) to form a connected graph, in terms of the numbers \( a_i \).

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