
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 a domain of \(\mathbf{R}\). When \( x < 0 \), \( f(x) > 1 \), and for any real numbers \( x \) and \( y \) in \(\mathbf{R}\), the equation \( f(x + y) = f(x) f(y) \) holds. The sequence \(\{a_n\}\) satisfies \( a_1 = f(0) \), and
\[ 
f\left( a_{n+1} \right) = \frac{1}{f\left( -2 - a_n \right)} \quad (n \in \mathbf{N}). 
\]

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