
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
The function \( f(x) \) defined on the set of real numbers \( \mathbf{R} \) satisfies \( f(x+1) = \frac{1+f(x+3)}{1-f(x+3)} \). Determine the value of \( f(1) \cdot f(2) \cdots f(2008) + 2009 \).

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