
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
It is given that the lengths of the sides of a triangle are 6, 8, and 10.

Given that \( f\left(x+\frac{1}{x}\right)=x^3+\frac{1}{x^3} \) and \( f(4)=b \), find the value of \( b \).

Given that \( 2002^2 - 2001^2 + 2000^2 - 1999^2 + \ldots + 4^2 - 3^2 + 2^2 - 1^2 = c \), find the value of \( c \).

\( PQRS \) is a square, \( PTU \) is an isosceles triangle, and \( \angle TPU = 30^\circ \). Points \( T \) and \( U \) lie on \( QR \) and \( RS \) respectively. The area of \( \triangle PTU \) is 1. If the area of \( PQRS \) is \( d \), find the value of \( d \).

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