A Modified Collatz sequence

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 $a_1\gt 1$ that begins with the sequence DdDddUUdDD.

What is the smallest $a_1\gt 10^{15}$ that begins with the sequence "UDDDUdddDDUDDddDdDddDDUDDdUUDd"?