
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 $a\in\mathbb{R}$, let the functions $m(x)=x^2$ and $n(x)=a\ln(x+2)$.  
(Ⅰ) Define the function $f(x)= \begin{cases} m(x), & x\leq 0 \\ n(x), & x>0 \end{cases}$. If there exist two points A and B on the graph of function $f(x)$ such that $OA\perpendicular OB$ (O is the origin) and the midpoint of line segment AB is on the y-axis, find the set of values for $a$.  
(Ⅱ) If the function $g(x)=m(x)+n(x)$ has two extreme points $x_1$ and $x_2$, find the range of $g(x_1)+g(x_2)$.

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