
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 the points $(3,1)$ and $(-4,6)$ are on opposite sides of the line $3x-2y+a=0$, then the range of values for $a$ is _______.

(2) Given two points $A(-1,1)$ and $B(3,0)$, the unit vector in the same direction as $\overrightarrow{AB}$ is _______.

(3) Observe the following equations:

$1=1$

$2+3+4=9$

$3+4+5+6+7=25$

$4+5+6+7+8+9+10=49$

...

Following this pattern, the $n$th equation is _______.

(4) Given the set $M=\{1,2,3,4,5\}$, $N=\{(a,b)|a\in M, b\in M\}$, $A$ is any point in set $N$, $O$ is the origin of coordinates, then the probability that the line $OA$ intersects with $y=x^{2}+1$ is _______.

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