
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
Color the positive integers \( 1, 2, \ldots, 15 \) either blue or red such that:
1. The integer 15 is red;
2. If two different integers \( x \) and \( y \) are of different colors, and \( x + y \leq 15 \), then \( x + y \) is blue;
3. If two different integers \( x \) and \( y \) are of different colors, and \( xy \leq 15 \), then \( xy \) is red.

Determine the number of such coloring methods.

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