
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
(1) If $x=2$ is an extremum point of the function $f(x)=ax+ \frac{1}{x}$, then the value of the real number $a$ is ______.

(2) If the circle $x^{2}+y^{2}=4$ and the circle $x^{2}+y^{2}+2ay-6=0$ ($a > 0$) have a common chord of length $2 \sqrt{3}$, then $a=$ ______.

(3) If the function $f(x)=x^{2}- \frac{1}{2}\ln x+1$ has an extremum in a subinterval $(a-1,a+1)$ within its domain, then the range of the real number $a$ is ______.

(4) Given the ellipse $C$ with the equation $\frac{x^{2}}{4}+ \frac{y^{2}}{3}=1$, $A$, $B$ are the left and right vertices of the ellipse $C$, $P$ is a point on the ellipse $C$ different from $A$, $B$. The line $x=4$ intersects lines $PA$, $PB$ at points $M$, $N$ respectively. If $D(7,0)$, then the circle passing through points $D$, $M$, $N$ must also pass through a fixed point on the $x$-axis other than point $D$, and its coordinates are ______.

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