
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) Point $P$ is any point on the curve $y=x^{2}-\ln x$. The minimum distance from point $P$ to the line $x-y-4=0$ is ______.

(2) If the tangent line to the curve $y=g(x)$ at the point $(1,g(1))$ is $y=2x+1$, then the equation of the tangent line to the curve $f(x)=g(x)+\ln x$ at the point $(1,f(1))$ is ______.

(3) Given that the distance from point $P(1,0)$ to one of the asymptotes of the hyperbola $C: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 (a > 0, b > 0)$ is $\frac{1}{2}$, the eccentricity of the hyperbola $C$ is ______.

(4) A line passing through point $M(1,1)$ with a slope of $-\frac{1}{2}$ intersects the ellipse $C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 (a > b > 0)$ at points $A$ and $B$. If $M$ is the midpoint of segment $AB$, then the eccentricity of the ellipse $C$ is ______.

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