Explaining the shape of the horizontal range
In the very short run, the AS curve is perfectly price-elastic (i.e. on the diagram, it is a horizontal line). It is also referred to as the Keynesian range. In this time period, firms respond to a rise in demand for their product without considering the effects of the rising demand, such as higher prices. This can be generalised for most firms, but not all.
We can look at the example of a book shop or a food stall. It is highly unlikely for these business owners to revise their prices on daily basis based on their past sales records. Prices are more likely reviewed on a yearly basis, or at some longer time intervals. Regular price revision could complicate simple business accounting and may also dispel customers, especially if your price revisions keep increasing the price.
There are some firms that are able to change prices based on instantaneous demand. These would include the ride-hailing service providers, such as Uber and Grab. Their computer algorithms are programmed to respond their market conditions of the moment and price according to the level of demand at that instant. Firms that are able to do this are more of an exception rather than the norm.
Another way to view it would be from the resource employment viewpoint. In this time period, the real national output/income is much lower than the full employment level. Thus, there is an abundance of un-utilised or under-utilised resources. Should there be an increase in aggregate demand, the spare capacity will allow producers to increase output production easily without incurring high costs. Hence, there is no pressure on the general price level to increase.
Explaining the shape of the vertical range
In the long run, prices do not keep rising. The AS curve is simply perfectly price-inelastic (i.e. on the diagram, it is a vertical line). This is also called the classical range of the AS curve. In the LR, all firms in the economy reach their full employment level (i.e. resources have been fully-utilised to their maximum capacity) and there is no demand-deficient unemployment.
Price levels are fully flexible as all firms will make adjustments to their prices in response to changes in demand.
This is what the vertical line indicates, not what you have mentioned.
The model I have learnt for the AS curve is a hybrid model, which is also depicted by the Wiki page for AS. The "short run" here essentially corresponds to your so-called "very short run". The "long run" here is the same as yours.