
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 \( x, y, z, \) and \( w \) be positive real numbers such that
\[
\begin{array}{c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c}
x^2+y^2 &=& z^2+w^2 &=& 4050, \\
xz &=& yw &=& 2040.
\end{array}
\]
If \( T = x+y+z+w \), compute the value of \( \lfloor T \rfloor \).

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