
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 five points \[(0,0), \;(4,0),\;(1,3),\;(3,3),\; \left(-1,\frac{3}{2}\right)\], where the first four points form a trapezoid (bases parallel to the x-axis) and the final point lies along the line that bisects the trapezoid vertically, find the length of the major axis of the unique ellipse passing through these points, assuming the axes of the ellipse are parallel to the coordinate axes.

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