
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 the right triangle \(ABC\) (\(\angle C\) is a right angle), a point \(D\) is taken on the side \(BC\) such that \(\angle CAD=30^\circ\). From point \(D\), a perpendicular \(DE\) is dropped onto \(AB\). Find the distance between the midpoints of segments \(CE\) and \(AD\), given that \(AC=3\sqrt{3}\) and \(DB=4\).

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