
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 functions $f(x) = 2x^3 + ax$ and $g(x) = bx^2 + c$ whose graphs both pass through the point $P(2, 0)$ and have the same tangent line at point $P$,

(I) Determine the values of the real numbers $a$, $b$, and $c$;  
(II) Define the function $F(x) = f(x) + g(x)$ and determine the intervals of monotonicity for the function $F(x)$.

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