
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 country, there are some two-way roads between the cities. There are  $2010$  roads connected to the capital city. For all cities different from the capital city, there are less than  $2010$  roads connected to that city. For two cities, if there are the same number of roads connected to these cities, then this number is even.  $k$  roads connected to the capital city will be deleted. It is wanted that whatever the road network is, if we can reach from one city to another at the beginning, then we can reach after the deleting process also. Find the maximum value of  $k.$ 

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