I need a probability density function on $\mathbb{R}[0, 1]$ with certain properties.

Question

I'm searching for a very specific probability density function. It needs to have the following 6 properties:

1. Has a support of $\mathbb{R}[0, 1]$, i.e. $\int_{0}^{1}f(x)dx = 1$
2. The pdf must be analytic on the domain $[0,1]$.
3. Is both unimodal and single-peaked.
4. Must have a parameter for the mean which can be set anywhere between $0$ and $1$.
5. Must be monotonically increasing or from $0$ to the mean, then monotonically decreasing from the mean to $1$, i.e. the graph of the function cannot have valleys. It should look like an asymmetric bell-curve.
6. The mean and mode of the pdf are the same.

I've been able to think of probability density functions that satisfy at most 5 of these criteria, but never all 6 at once. For example, a bump function could satisfy all but criterion #2. The beta distribution could satisfy all but #6. The normal distribution could satisfy all but #1.

I need some help. Does such a probability density function exist, and if so, what is it?