gold_Moo_Route — USACO January_Contest
Task type: batch | Time limit: 2 s | Memory limit: 256 MB | Difficulty: 19
Algorithms: number theory, mathematics, fermat theorem, combinatorics, modular arithmetic

Farmer Nhoj dropped Bessie in the middle of nowhere! At time $t=0$, Bessie is
located at $x=0$ on an infinite number line. She frantically searches for an
exit by moving left or right by $1$ unit each second. However, there actually is
no exit and after $T$ seconds, Bessie is back at $x=0$, tired and resigned.
Farmer Nhoj tries to track Bessie but only knows how many times Bessie crosses
$x=.5, 1.5, 2.5, \ldots, (N-1).5$, given by an array $A_0,A_1,\dots,A_{N-1}$
($1\leq N \leq 10^5$, $1 \leq A_i \leq 10^6$). Bessie never reaches $x>N$ nor
$x<0$.
In particular, Bessie's route can be represented by a string of
$T = \sum_{i=0}^{N-1} A_i$ $L$s and $R$s where the $i$th character represents
the direction Bessie moves in during the $i$th second. The number of direction
changes is defined as the number of occurrences of  $LR$s plus the number of
occurrences of $RL$s.
Please help Farmer Nhoj count the number of routes Bessie could have taken that
are consistent with $A$ and minimize the number of direction changes. It is
guaranteed that there is at least one valid route.
INPUT FORMAT (input arrives from the terminal / stdin):
The first line contains $N$. The second line contains $A_0,A_1,\dots,A_{N-1}$.
OUTPUT FORMAT (print output to the terminal / stdout):
The number of routes Bessie could have taken, modulo $10^9+7$.
SAMPLE INPUT:
2
4 6
SAMPLE OUTPUT:
2
Bessie must change direction at least 5 times. There are two routes 
corresponding to Bessie changing direction exactly 5 times:
RRLRLLRRLL
RRLLRRLRLL
SCORING:
Inputs 2-4: $N\le 2$ and $\max(A_i)\le 10^3$
Inputs 5-7: $N\le 2$
Inputs 8-11: $\max(A_i)\le 10^3$
Inputs 12-21: No additional constraints.
Problem credits: Brandon Wang, Claire Zhang, and Benjamin Qi

Starter code: