
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
For a $1 \times n$ rectangular grid, where the cells are labeled sequentially from left to right as $1, 2, \ldots, n$, each cell can be colored using one of three colors: 1, 2, or 3. The coloring must satisfy the following conditions:
1. Even-numbered cells can be colored with any of the three colors.
2. Odd-numbered cells can only be colored with either color 1 or color 3.
3. Adjacent cells must have different colors.

How many possible ways are there to color this rectangular grid?

(This is from the 43rd Austrian Mathematics Competition)

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