
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
Given that $y$ is directly proportional to $x-1$, and when $x=-1$, $y=4$.  
$(1)$ Find the analytical expression of $y$ in terms of $x$;  
$(2)$ Points $M(x_{1}, y_{1})$ and $N(x_{2}, y_{2})$ lie on the graph of the function in $(1)$, if $x_{1} > x_{2}$, then $y_{1}$ ______ $y_{2}$ (Fill in "$>$", "$=$", or "$<$");  
$(3)$ The graph of the function in $(1)$ is shifted down by $4$ units, and the new graph intersects the $x$-axis and $y$-axis at points $A$ and $B$ respectively. Find the area of $\triangle AOB$.

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