
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
A function \( f(x) \) defined on the set of real numbers \(\mathbf{R}\) satisfies the following three conditions:
1. \( f(x) > 0 \) when \( x > 0 \),
2. \( f(1) = 2 \),
3. For any \( m, n \in \mathbf{R} \), \( f(m+n) = f(m) + f(n) \).

Let the sets be defined as:
\[ 
A = \left\{ (x, y) \mid f(3x^2) + f(4y^2) \leq 24 \right\},
\]
\[ 
B = \{ (x, y) \mid f(x) - f(ay) + f(3) = 0 \},
\]
\[ 
C = \left\{ (x, y) \mid f(x) = \frac{1}{2} f(y^2) + f(a) \right\}.
\]

If \( A \cap B \neq \varnothing \) and \( A \cap C \neq \varnothing \), determine the range of the real number \( a \).

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