
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 diagram below, \(ABCD\) is a rectangle and \(E\) is the foot of the perpendicular dropped from \(A\) to the diagonal \(BD\). The distances from point \(E\) to the sides \(DC\), \(BC\), and \(AB\) are \(n\), \(1\), and \(x\), respectively. Let \(d\) be the length of the diagonal \(BD\).

a) Verify that \(DE = x^{2} \sqrt{1 + x^{2}}\).

b) Verify that \(n = x^{3}\).

c) Verify that \(d^{2/3} - x^{2/3} = 1\).

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