
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) The coordinates of the focus of the parabola $4x^2=y$ are _______.

(2) The formula for the distance $d$ from a point $(x_0,y_0)$ to the line $Ax+By+C=0$ is $d= \frac{|Ax_0+By_0+C|}{\sqrt{A^2+B^2}}$. By analogy, the distance from the point $(0,1,3)$ to the plane $x+2y+3z+3=0$ is _______.

(3) The equation of the hyperbola that has the same asymptotes as $\frac{x^2}{2} -y^2=1$ and passes through $(2,0)$ is _______.

(4) Given that the eccentricity of the ellipse $\frac{x^2}{a^2}+ \frac{y^2}{b^2}=1$ $(a > b > 0)$ is $\frac{1}{2}$, $A$ and $B$ are the left and right vertices of the ellipse, respectively, and $P$ is a point on the ellipse different from $A$ and $B$. The angles of inclination of lines $PA$ and $PB$ are $\alpha$ and $\beta$, respectively. Then $\frac{\cos (\alpha +\beta )}{\cos (\alpha -\beta )}=$ _______.

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