
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 an equilateral triangle with side length \( n \), divided into unit triangles as illustrated, let \( f(n) \) be the number of paths from the top-row triangle to the triangle in the center of the bottom row. The path must move through adjacent triangles sharing a common edge, never revisiting any triangle, and never moving upwards (from a lower row to an upper row). An example path is illustrated for \( n = 5 \). Determine the value of \( f(2012) \).

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