
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 \(a, b, c\), and \(d\) be positive real numbers such that
\[
\begin{array}{c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c}
a^2+b^2&=&c^2+d^2&=&2500,\\
ac+bd&=&1500.
\end{array}
\]
If \(S=a+b+c+d\), compute the value of \(\lfloor S\rfloor\).

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