For some positive integers $k$, there exists an integer partition of the form $4^t=2^t+k$,
where $4^t$, $2^t$, and $k$ are all positive integers and $t$ is a real number.

The first two such partitions are $4^1=2^1+2$ and $4^{1.5849625\cdots }=2^{1.5849625\cdots }+6$.

Partitions where $t$ is also an integer are called perfect.
For any $m\ge 1$ let $P(m)$ be the proportion of such partitions that are perfect with $k\le m$.
Thus $P(6)=1/2$.

In the following table are listed some values of $P(m)$.

$$\begin{array}{rl}P(5)& =1/1\\ P(10)& =1/2\\ P(15)& =2/3\\ P(20)& =1/2\\ P(25)& =1/2\\ P(30)& =2/5\\ \cdots & \\ P(180)& =1/4\\ P(185)& =3/13\end{array}$$

Find the smallest $m$ for which $P(m)<1/12345$.