
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 \( f(x) \) with the domain of all 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 following sets be defined as:
\[ A = \{ (x, y) \mid f(3x^2) + f(4y^2) \leq 24 \} \]
\[ B = \{ (x, y) \mid f(x) - f(ay) + f(3) = 0 \} \]
\[ C = \{ (x, y) \mid f(x) = \frac{1}{2} f(y^2) + f(a) \} \]

Given that \( A \cap B \neq \varnothing \) and \( A \cap C \neq \varnothing \), find the range of the real number \( a \).

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