,
,
cut three common chords concurrent at 
.
,
which are internally tangent to all 
.
be a parallelogram of center 
.
,
and 
such that 
.
be given . On the extension of the segment 
beyond 
,
is chosen arbitrarily. Let the line through 
at the point 
.
be the intersection of the lines 
and 
.
is tangent to the circumcircle of the triangle 
.
Y by
Sabbir noticed one day that everyone in the city of BdMO has a distinct word of length 
, where each letter is either 
or 
. Sabbir saw that two citizens are friends if one of their words can be altered a few times using a special rule and transformed into the other ones word. The rule is, if somewhere in the word 
is located consecutively, then these letters can be changed to 
or if is located somewhere in the word consecutively, then these letters can be changed to (if wanted, the word can be kept as it is, without making this change.) For example 
can be transformed into 
(the opposite is also possible.) Now Sabbir made a team of 
citizens where no one is friends with anyone. What is the highest value of 
, where each letter is either 
or 
. Sabbir saw that two citizens are friends if one of their words can be altered a few times using a special rule and transformed into the other ones word. The rule is, if somewhere in the word 
is located consecutively, then these letters can be changed to 
or if is located somewhere in the word consecutively, then these letters can be changed to (if wanted, the word can be kept as it is, without making this change.) For example 
can be transformed into 
(the opposite is also possible.) Now Sabbir made a team of 
citizens where no one is friends with anyone. What is the highest value of 
