
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
The lengths of the non-parallel sides of a symmetrical trapezoid are $5 \mathrm{~cm}$, and the difference between the lengths of the parallel sides is $6 \mathrm{~cm}$. If the length of the base of the trapezoid is reduced by $10 \mathrm{~cm}$ and the height is increased by $1 \mathrm{~cm}$, the area of the trapezoid remains unchanged. What are the lengths of the sides of the trapezoid? What are the height and the area of the trapezoid?

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