
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
Each of the cells of a  $7 \times 7$  grid is painted with a color chosen randomly and independently from a set of  $N$  fixed colors. Call an edge hidden if it is shared by two adjacent cells in the grid that are painted the same color. Determine the least  $N$  such that the expected number of hidden edges is less than  $3$ .

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