
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 functions \( u \) and \( v \) of the independent variables \( x \) and \( y \) are given by the system of equations

\[ 
\begin{cases} 
x u - y v - 1 = 0 \\ 
x - y + u - v = 0 
\end{cases} 
\]

Find \( du, dv, u_{x}^{\prime}, u_{y}^{\prime}, v_{x}^{\prime}, v_{y}^{\prime}, d^{2}u \).

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