
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 country of Squareland is shaped like a square and is divided into 64 congruent square cities. We want to divide Squareland into states and assign to each state a capital city so that the following rules are satisfied:

(a) Every city lies entirely within one state.

(b) Given any two states, the numbers of cities in them differ by at most 1.

(c) Any city in a state shares at least one corner with the state's capital.

What is the smallest possible number of states?

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