
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
There are exactly \( N \) values of \( \alpha \) satisfying the equation \( \cos^3 \alpha - \cos \alpha = 0 \), where \( 0^\circ \leq \alpha \leq 360^\circ \). Find \( N \).

If the \( N \)-th day of May in a year is Thursday and the \( K \)-th day of May in the same year is Monday, where \( 10 < K < 20 \), find \( K \).

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