
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) Given a geometric sequence $\{a_n\}$ satisfying $a_1+a_3=10$ and $a_2+a_4=5$, find $a_8=$ ______.

(2) Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ with an angle of $\frac{\pi}{6}$ between them, $|\overrightarrow{a}|=2$, and $|\overrightarrow{b}|=\sqrt{3}$, find $\overrightarrow{a}\cdot(2\overrightarrow{b}-\overrightarrow{a})=$ ______.

(3) Find the length of the chord intercepted on the circle $(x-6)^2+(y-\sqrt{2})^2=12$ by the line passing through the point $(1,0)$ and parallel to the line $x-\sqrt{2}y+3=0$.

(4) In an acute triangle $ABC$, where the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, if $a=4$, $b=5$, and the area of $\triangle ABC$ is $5\sqrt{3}$, find $c=$ ______; $\sin A=$ ______.

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