
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
Given \( n \) points \( P_1, P_2, \cdots, P_n \) arranged successively on a straight line, each point is colored with one of the colors white, red, green, blue, or purple. A coloring is considered "good" if for any two adjacent points \( P_i \) and \( P_{i+1} \) (for \( i = 1, 2, \cdots, n-1 \)), either these two points are the same color or at least one of them is white. Find the total number of good colorings.

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