
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
Given the function $f(x)$ defined on $\mathbf{R}$ satisfies $f(x+y)=f(x)+f(y)$, and for $x>0$, we have $f(x)<0$. Additionally, we know $f(1)=-2$. Let $a>0$ be a constant. Solve the inequality $\frac{1}{2} f\left(a x^{2}\right)-f(x)>\frac{1}{2} f\left(a^{2} x\right)-f(a)$.

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