
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 the function \( y = f(x) \) have the domain \( \mathbf{R} \). When \( x < 0 \), \( f(x) > 1 \). Moreover, for any real numbers \( x, y \in \mathbf{R} \), the equation \( f(x+y) = f(x) f(y) \) holds. A sequence \( \left\{a_n\right\} \) is defined such that \( a_1 = f(0) \) and \( f\left(a_{n+1}\right) = \frac{1}{f\left(-2 - a_n\right)} \) for \( n \in \mathbf{N} \).
1. Find the value of \( a_{2003} \).
2. If the inequality \( \left(1+\frac{1}{a_1}\right)\left(1+\frac{1}{a_2}\right) \cdots\left(1+\frac{1}{a_n}\right) \geqslant k \cdot \sqrt{2n+1} \) holds for all \( n \in \mathbf{N} \), determine the maximum value of \( k \).

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