
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 $f(x)=x^{2}+ax+3$, $g(x)=(6+a)\cdot 2^{x-1}$.

(I) If $f(1)=f(3)$, find the value of the real number $a$;

(II) Under the condition of (I), determine the monotonicity of the function $F(x)=\frac{2}{1+g(x)}$ and provide a proof;

(III) When $x \in [-2,2]$, $f(x) \geqslant a$, ($a \notin (-4,4)$) always holds, find the minimum value of the real number $a$.

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