
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
If \( f(x) \) and \( g(x) \) are both odd functions, and \( F(x) = a f(x) + b g(x) + 2 \), which attains a maximum value of 8 on \( (0, +\infty) \), determine the value that \( F(x) \) attains on \( (-\infty, 0) \).

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