
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 that functions f(x) and g(x) are functions defined on $\mathbb{R}$, where f(x) is an odd function and g(x) is an even function, and 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}}} > -2, determine the possible values of the real number a.

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