
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) = ax^2 + bx + 1 (where a and b are real numbers), x ∈ R, and F(x) = $$\begin{cases} f(x), & x>0 \\ -f(x), & x<0 \end{cases}$$, suppose m > 0, n < 0, m + n > 0, a > 0, and f(x) is an even function. Determine whether F(m) + F(n) can be greater than zero.

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