
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 rectangle $EFGH$, $J$ is a point on $FG$ such that $\angle EJH=90^{\circ}$. $UV$ is perpendicular to $FG$ with $FU=UJ$ as shown. $JH$ intersects $UV$ at $K$. Point $L$ is on $GH$ such that $LE$ passes through $K$. In $\triangle JKE$, $JE=15$, $EK=20$ and $JK=25$. Determine the lengths of $FJ$ and $UK$.

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