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To solve the given problem, let's break it down step by step:

1. **Given Information:**
   - \( f(x) \) is a differentiable function on \( \mathbb{R} \).
   - For \( x \geq 0 \), \( f'(x) > f'(-x) \).
   - \( g(x) = f(x) + f(-x) \).
   - For any \( x \) in the interval \( \left[\frac{1}{2}, 1\right] \), the inequality \( g(ax + 1) \leq g(x - 2) \) holds.

2. **Objective:**
   - Determine the possible values of \( a \).

3. **Analysis of \( g(x) \):**
   - Since \( g(x) = f(x) + f(-x) \), we need to check its behavior under transformation.
   - Given \( f'(x) > f'(-x) \) for \( x \geq 0 \), it indicates that \( f(x) \) is increasing faster than \( f(-x) \) when \( x \geq 0 \).

4. **Applying the inequality:**
   - We need to analyze \( g(ax + 1) \) and \( g(x - 2) \).
   - Let \( h(x) = g(x) \) and see how \( h(ax + 1) \) compares to \( h(x - 2) \).

5. **Testing values:**
   - Let's consider \( x = \frac{1}{2} \) and \( x = 1 \):
     - If \( x = \frac{1}{2} \):
       \[
       g\left(a \cdot \frac{1}{2} + 1\right) \leq g\left(\frac{1}{2} - 2\right)
       \]
       \[
       g\left(\frac{a}{2} + 1\right) \leq g\left(-\frac{3}{2}\right)
       \]

     - If \( x = 1 \):
       \[
       g(ax + 1) \leq g(1 - 2)
       \]
       \[
       g(a + 1) \leq g(-1)
       \]

6. **Considering \( a = -1 \):**
   - For \( a = -1 \), \( \frac{a}{2} + 1 = \frac{-1}{2} + 1 = \frac{1}{2} \).
   - So, for \( x = \frac{1}{2} \):
     \[
     g\left(\frac{1}{2}\right) \leq g\left(-\frac{3}{2}\right)
     \]
   - For \( x = 1 \):
     \[
     g(0) \leq g(-1)
     \]

7. **Verification of choices:**
   - \( a = -1 \) satisfies the inequality because it maps the left endpoint of the interval to the same endpoint.
   - Other values like \( a = 0, 1, 2 \) do not satisfy as they map to out-of-bound intervals or non-symmetric points.

Thus, the possible value for \( a \) is \( \boxed{-1} \).

Would you like further details on any specific part of the problem, or do you have additional questions?

Here are some questions you might consider asking next:
1. Can you explain how \( g(x) = f(x) + f(-x) \) behaves for different \( f(x) \)?
2. How does differentiability affect the inequality \( g(ax + 1) \leq g(x - 2) \)?
3. Why are transformations like \( ax + 1 \) important in function inequalities?
4. How can we use derivatives to analyze inequalities in functions?
5. What other values for \( a \) might we test in similar problems?

**Tip:** When dealing with inequalities involving transformations, always consider the impact of the function's derivative and how it affects the function's behavior on different intervals.

Explore how to solve function inequalities involving differentiable functions step-by-step. This problem focuses on analyzing the behavior of g(x) = f(x) + f(-x) under different transformations and inequalities. Suitable for advanced high school students.

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