
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) There are 10 numbers that can form a geometric sequence with the first term being 1 and the common ratio being -3. If one number is randomly selected from these 10 numbers, the probability that it is less than 8 is ______.

(2) Suppose $(5,4)$ and $(5,5)$ are the two foci of the hyperbola $(5,6)$, and point $(6,1)$ is on the hyperbola and satisfies $(6,2)$, then the area of $(6,3)$ is ______.

(3) It is known that the circle $|\frac{m}{2}-1|=\sqrt{\frac{1+{m}^{2}}{4}}$ is tangent to the directrix of the parabola ${{y}^{2}}=4x$, then $\frac{3}{4}$ ______.

(4) Let ${e}_{1},{e}_{2}$ be the eccentricities of an ellipse and a hyperbola, respectively, both having common foci ${F}_{1},{F}_{2}$. If $P$ is a common point of the two curves, and it satisfies $| \overrightarrow{PF_{1}}+ \overrightarrow{PF_{2}}|=| \overrightarrow{{F}_{1}{F}_{2}}|$, then the value of $ \frac{{e}_{1}{e}_{2}}{ \sqrt{{e}_{1}^{2}+{e}_{2}^{2}}}$ is ______.

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