
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
Let \( p, q, r, \) and \( s \) be positive real numbers such that 
\[
\begin{array}{c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c}
p^2+q^2 &=& r^2+s^2 &=& 4016, \\
pr &=& qs &=& 2000.
\end{array}
\]
If \( T = p+q+r+s\), compute the value of \(\lfloor T \rfloor\).

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