
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) If point $M$ is on the circle $(x-5)^{2}+(y-3)^{2}=9$, then the shortest distance from point $M$ to the line $3x+4y-2=0$ is ______.

(2) Given that the mean of the sample data $x_{1}$, $x_{2}$, …, $x_{n}$ is $\overline{x}=5$, then the mean of the sample data $2x_{1}+1$, $2x_{2}+1$, …, $2x_{n}+1$ is ______.

(3) Players A and B play a guessing game. First, A thinks of a number denoted as $a$, then B guesses the number A thought of, denoted as $b$, where $a$, $b∈\{0,1,2,…,9\}$. If $|a-b|≤ 1$, then A and B are considered to have a "telepathic connection". If two people are randomly chosen to play this game, the probability that they have a "telepathic connection" is ______.

(4) Let $\triangle ABC$ have sides opposite to angles $A$, $B$, and $C$ denoted as $a$, $b$, and $c$ respectively, and given that $a=2$, $\cos C=-\frac{1}{4}$, and $3\sin A=2\sin B$, then $c=$ ______.

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