\problemname{Bubbly Troubly}

You may have seen a champagne tower at a wedding or an exclusive Hollywood A-list party.
In a typical three-level tower, the first (lowest) level contains 9 glasses touching in a square
pattern. The second level contains 4 glasses touching in a square pattern centered
above the first level, and the third level contains 1 glass centered above the other levels. Figure A.1 shows a a top-down view of this tower.
\begin{center}
\scalebox{0.6}{\includegraphics{threeleveltower}}\\
Figure A.1
\end{center}

Champagne is always poured directly into the top glass. In this example,
once the top glass fills and starts to overflow, it {\em immediately} begins filling the 4 glasses below
(i.e., assume overflowing champagne travels instantaneously to any glasses below).
Once the 4 glasses on
the second level fill, they begin overflowing to the 9 on the bottom level.
Note that in this example the 4 on the second level finish filling at the same time, but
the 9 on the lowest level finish filling at different times. This means
that there will be some amount of spilled champagne before the tower has
finished filling -- this is an acceptable price to pay for such a beautiful
sight.

The new fad is to make interesting patterns or imagery out of champagne glasses.
These new-fangled ``towers" needn't appear structurally sound; they can be held
in place with complex support systems designed so as not to interfere with the
overflowing champagne. Each of these new towers will always have a single
highest glass into which all champagne is directly poured.

If two glass rims coincide vertically (i.e., have the same center and radius), then  no accumulation occurs into the lower glass from the upper glass (though the overflow champagne from the upper glass may still be collected by other lower glasses).
%If champagne overflows onto the rim of another glass {\it exactly}, then no accumulation occurs into that glass from that champagne (though that champagne can still be collected by lower glasses).
Additionally, a single point of champagne overflow causes no
measurable accumulation.  In other words, measurable accumulation only occurs when a
non-point arc of champagne overflows to the interior of a glass.

Your task is to determine whether a proposed champagne tower will fill to completion, and if so, how long it will take.


\section*{Input}
The input begins with a single integer $n$ representing the number of champagne glasses
in the tower ($1 \leq n \leq 20$). The next $n$ lines each describe a champagne glass. Each glass description consists of 5 values $x$ $y$ $z$ $r$ $v$ with ($x$, $y$, $z$) representing
the center of the glass's rim ($0 \leq x, y \leq 1000; 1 \leq z \leq 1000$), $r$
representing its radius ($1 \leq r \leq 1000$), and $v$ representing its volume measured
in milliliters ($1 \leq v \leq 1000$). All input values are integers, and the top glass
is filled at a constant 100 milliliters per second.


\section*{Output}
Display the number of seconds after which the tower will be completely filled,
or {\tt Invalid} if the proposed champagne tower will never fill completely.
Round answers to the hundredths place. Always print answers to two decimal
places and include the leading 0 on answers between 0 and 1.
Output values will always be $\leq 10^6$ seconds (or 11 days, 13 hours, 46 minutes and 40 seconds, whichever you prefer).

