Dr$.$ Eva Thomas is an underwater geographer studying mountains and valleys in the Atlantic Ocean. She models mountain and valley ranges using integers whose digits alternate between increasing and decreasing. You have been hired to write a program that, given two integers a and b$,$ determines how many numbers model mountain and valley ranges in the range between a and b$,$ inclusive.

An n$-$digit integer, for n $>=2,$ with digits d$1$d$2$d$3$d$4...$dn models:

a mountain range if it has the pattern $/\backslash /...,$ that is$,$ d$1<$ d$2$ d$2,>$ d$3,$ d$3<$ d$4,$ etc.

a valley range if it has the pattern $\backslash /\backslash ...,$ that is d$1>$ d$2,$ d$2<$ d$3,$ d$3>$ d$4,$ etc.

Examples:

$1503$ has $1<5,5>0,$ and $0<3,$ so it models a mountain range.

$7120$ has $7>1,1<2,$ and $2>0,$ so it models a valley range.

$7113$ has consecutive $1$s$,$ so it models neither.

For a $=15$ and b $=25,$ the answer is $8$ mountain ranges and $2$ valley ranges.

The numbers $15,16,17,18,19,23,24,$ and $25$ are mountain ranges.

The numbers $20$ and $21$ are valley ranges.

The number $22$ is neither a mountain nor a valley range.

For a $=1234$ and b $=4321,$ the answer is $753$ mountain ranges and $351$ valley ranges.

For a $=10$ and b $=1000,$ the answer is $276$ mountain ranges and $330$ valley ranges.

See the following how we got the solution for a $=15$ and b $=25,$ and how to handle $2,3,$ or $4$ digits. These are examples of some of the cases; your program will need to account for more digits.