
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
A country has $k$ provinces, and each province has $a_{1}, a_{2}, \cdots, a_{k}$ cities respectively. The plan is to establish some (bidirectional) flight routes between these cities, ensuring that any city can be reached from any other city through one or more flights. No flight routes are to be set within the same province, and the total number of flight routes should be minimized. How many different methods are there to design these flight routes?

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