
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
1. A quadratic function \( f(x) = ax^2 + bx + c \) (where \( a, b, c \in \mathbb{R} \) and \( a \neq 0 \)) satisfies:
   1. For all \( x \in \mathbb{R} \), \( 4x \leq f(x) \leq \frac{1}{2}(x+2)^2 \) always holds.
   2. \( f(-4 + 2\sqrt{3}) = 0 \). Find \( f(x) \).

2. Given \( f_1(x) = \frac{3}{2+x} \) and \( f_{n+1}(x) = f_1(f_n(x)) \), find \( f_{2009}(0) \).

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