
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
Perpendiculars are drawn from the vertex of an acute angle of a rhombus to the lines containing the sides of the rhombus which do not include this vertex. The length of each perpendicular is 3 cm and the distance between their bases is \( 3 \sqrt{3} \) cm. Calculate the lengths of the diagonals of the rhombus.

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