
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
As shown in the diagram, \(ABCD\) is a square with side length \(6\). \(ADGH\) is a trapezoid, and points \(E\) and \(F\) are the midpoints of \(AD\) and \(GH\), respectively. It is given that \(HF = 6\), \(EF = 4\), and \(EF \perp GH\). Line segment \(HE\) is extended to intersect \(CD\) at point \(I\). A line segment \(IJ\) is drawn perpendicular to \(HA\). Find the length of \(IJ\).

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