
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 domain of the function \( f(x) \) is the set of real numbers \(\mathbf{R}\). Given that for \( x > 0 \), \( f(x) > 0 \), and for any \( m, n \in \mathbf{R} \), it holds that
\[ f(m+n) = f(m) + f(n). \]
(1) Discuss the parity and monotonicity of the function \( f(x) \);
(2) Let the sets be defined as
\[ 
\begin{array}{l}
A=\left\{(x, y) \mid f\left(3 x^{2}\right)+f\left(4 y^{2}\right) \leqslant 24\right\}, \\
B=\{(x, y) \mid f(x)-f(a y)+f(3)=0\}, \\
C=\left\{(x, y) \mid f(x)=\frac{1}{2} f\left(y^{2}\right)+f(a)\right\},
\end{array}
\]
and given that \( f(1)=2 \). If \( A \cap B \neq \varnothing \) and \( A \cap C \neq \varnothing \), find the range of the real number \( a \).

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