
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
Four different stamps form a $2 \times 2$ block. By tearing off 0 to 3 stamps from this block, you can obtain 13 different connected pieces of stamps. Determine the number of different connected pieces of stamps obtained from a $2 \times n$ block of $2n$ different stamps after tearing off 0 to $2n-1$ stamps.

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