
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
The set $A$ consists of functions $f(x)$ that have the following properties: 
1. The domain of the function $f(x)$ is $[0,+\infty)$.
2. The range of the function $f(x)$ is $[-2,4)$.
3. The function $f(x)$ is increasing on $[0,+\infty)$. 

Explore the following two questions:

(I) Determine whether the functions $f_1(x)=\log_2\left( \sqrt{x}+ \frac{1}{4}\right)$ and $f_2(x)=4-6\left( \frac{1}{2}\right)^x$ (for $x\geqslant 0$) belong to set $A$, and briefly explain the reasons.

(II) For the function $f(x)$ that you believe belongs to set $A$ from question (I), does the inequality $f(x)+f(x+2) < 2f(x+1)$ always hold for any $x\geqslant 0$? If it holds, please provide a proof; if not, explain why.

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