
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 the parabola intersects the $x$-axis at $A(-1,0)$, $B(3,0)$, and passes through point $C(0,3)$.  
$(1)$ Find the analytical expression of the parabola.  
$(2)$ Let the vertex of the parabola be $M$, find the area of $\triangle ABM$.

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