A modified Collatz sequence of integers is obtained from a starting value $a_1$ in the following way:
$a_{n + 1} = \frac{a_n}{3}$ if $a_n$ is divisible by 3. We shall denote this as a large downward step, “D”.
$a_{n + 1} = \frac{4a_n + 2}{3}$ if $a_n$ divided by 3 gives a remainder of 1. We shall denote this as an upward step, “U”.
$a_{n + 1} = \frac{2a_n - 1}{3}$ if $a_n$ divided by 3 gives a remainder of 2. We shall denote this as a small downward step, “d”.
The sequence terminates when some $a_n = 1$.
Given any integer, we can list out the sequence of steps. For instance if $a_1 = 231$, then the sequence $\{a_n\} = \{231, 77, 51, 17, 11, 7, 10, 14, 9, 3, 1\}$ corresponds to the steps “DdDddUUdDD”.
Of course, there are other sequences that begin with that same sequence “DdDddUUdDD…”.
For instance, if $a_1 = 1004064$, then the sequence is DdDddUUdDDDdUDUUUdDdUUDDDUdDD.
In fact, 1004064 is the smallest possible $a_1 > {10}^6$ that begins with the sequence DdDddUUdDD.
What is the smallest $a_1 > {10}^{15}$ that begins with the sequence “UDDDUdddDDUDDddDdDddDDUDDdUUDd”?