
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 focal distance of the parabola $4x^{2}=y$ is \_\_\_\_\_\_\_\_\_\_\_\_
2. The equation of the hyperbola that has the same asymptotes as the hyperbola $\frac{x^{2}}{2} -y^{2}=1$ and passes through $(2,0)$ is \_\_\_\_\_\_\_\_\_\_\_\_
3. In the plane, the distance formula between a point $(x_{0},y_{0})$ and a line $Ax+By+C=0$ is $d= \frac{|Ax_{0}+By_{0}+C|}{\sqrt{A^{2}+B^{2}}}$. By analogy, the distance between the point $(0,1,3)$ and the plane $x+2y+3z+3=0$ is \_\_\_\_\_\_\_\_\_\_\_\_
4. If point $A$ has coordinates $(1,1)$, $F_{1}$ is the lower focus of the ellipse $5y^{2}+9x^{2}=45$, and $P$ is a moving point on the ellipse, then the maximum value of $|PA|+|PF_{1}|$ is $M$, the minimum value is $N$, so $M-N=$ \_\_\_\_\_\_\_\_\_\_\_\_

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