
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
Consider a unit square where two edges are colored with color \( i \), and the other two edges are colored with two different colors other than \( i \). Such a unit square is called "dominant in color \( i \)". As shown in the diagram, a \( 1 \times 3 \) grid has a total of 10 unit length lines. We want to color these 10 lines with three colors: red, yellow, and blue, such that there is exactly one square dominant in red, one dominant in yellow, and one dominant in blue. Find the number of ways to color these lines. Provide the answer as a numerical value.

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