
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 \(\mathbf{R}\) satisfies: for any real numbers \( m \) and \( n \), \( f(m+n)=f(m) \cdot f(n) \), and for \( x>0 \), \( 0<f(x)<1 \).

(1) Determine the monotonicity of \( f(x) \).

(2) Let \( A=\left\{(x, y) \mid f\left(x^{2}\right) \cdot f\left(y^{2}\right)>f(1)\right\} \) and \( B=\{(x, y) ; f(a x-y+\sqrt{2})=1, a \in \mathbf{R}\} \). If \( A \cap B=\varnothing \), determine the range of the value of \( a \).

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