
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
Let $a_{1}, a_{2}, \cdots, a_{2002}$ be positive real numbers such that $\frac{1}{2+a_{1}}+\frac{1}{2+a_{2}}+\cdots+\frac{1}{2+a_{2002}}=\frac{1}{2}$. What is the minimum value of $a_{1} \cdot a_{2} \cdot a_{3} \cdots \cdots \cdot a_{2002}$?

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