
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
In the Cartesian coordinate system $xOy$, the graph of the function $y=kx+b\left(k\neq 0\right)$ passes through the points $A\left(-1,2\right)$ and $B\left(1,4\right)$.  
$(1)$ Find the analytical expression of this function;  
$(2)$ When $x \gt 2$, for each value of $x$, the value of the function $y=\frac{1}{2}x+n$ is less than the value of the function $y=kx+b\left(k\neq 0\right)$ and greater than $5$. Write down the value of $n$ directly.

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