
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 $f(x)$ and $g(x)$ are functions with domain $R$, where $f(x)$ is an odd function and $g(x)$ is an even function, satisfying $f(x) + g(x) = ax^{2} + x + 2$. If for any $1 < x_{1} < x_{2} < 2$, $\frac{{g(x_{1}) - g(x_{2})}}{{x_{1} - x_{2}}} > -3$ holds, find the range of the real number $a$.

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