
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 $n$ points $P_{1}, P_{2}, \cdots, P_{n}$ arranged sequentially on a straight line, each of which is colored in one of the colors white, red, green, blue, or purple. If for any two adjacent points $P_{i}$ and $P_{i+1}$ $(i=1,2, \cdots, n-1)$, either they are the same color, or at least one of them is white, then the coloring is called a good coloring. Find the total number of good colorings.

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