When connected to ideal voltage sources, then assume that they can sink and provide enough current to maintain their voltages. That means \$V_{B}\$ is \$V_{IN}\$ and \$V_{E}\$ is GND. So \$V_{BE}\$ is simply \$V_{IN}\$.
Now for \$V_{OUT}\$, which \$V_{CE}\$, which is of course not the same as \$V_{CESAT}\$, which is typical \$V_{CE}\$ at saturation.
$$V_{CESAT}$$
$$if \quad B_{F}(V_{IN} - V_{BESAT})/((B_{F} +1)(R_{PB} + R_{PE})) > (V_{IN} - V_{BESAT} + V_{CESAT})/(R_{C} + R_{PC});$$
$$V_{DD} - (R_{C}+R_{PC})B_{F}(V_{IN} - V_{BESAT})/((B_{F} +1)(R_{PB} + R_{PE}))$$
$$if \quad otherwise$$
\$R_{PB}\$ is parasitic base resistance, \$R_{PE}\$ is parasitic emitter resistance and \$R_{PC}\$ parasitice collector resistance. These are usually 0.5-1.5 ohms, so only useful to take into account when BJT is wrking more in current mode. Such as now that base and emitter are bot connected directly to sources.
For even more accurate calculation, the \$V_{CESAT}\$ and \$V_{BESAT}\$ are just the minimum and maximum boundaries, taken as a single entity, that actual \$V_{CE}\$ and \$V_{BE}\$ can take at saturation given the current formulas from Ebers-Moll or Gummel-Poon models. This may not be as accurate as starting with said 2 previous models and using Convergence, Numerical Methods or Symbolic Engines, but this is good enough to determine the regions and modes the transistors are in.
EDIT:
By Ebers-Moll model, I do not mean the regular this:
https://wikimedia.org/api/rest_v1/media/math/render/svg/e95565b2aa76041d3124461d12091effe1afe96a
I meant the complete:
https://wikimedia.org/api/rest_v1/media/math/render/svg/4062fe7275c023cf696f4be157c3725d95299b07
Couldn't attach the images as they are SVG. I won't go to the trouble of typing it, either.