Why do all states appear identical under the function approximation in the Short Corridor task? This is the Short Corridor problem taken from the Sutton &amp; Barto book. Here it's written:

The problem is difficult because all the states appear identical under the function approximation

But this doesn't make much sense as we can always choose states as 0,1,2 and corresponding feature vectors as
x(S = 0,right) = [1  0  0  0  0  0]
x(S = 0 , left)  = [0  1  0  0  0  0]
x(S = 1,right) = [0  0  1  0  0  0]
x(S = 1 , left) = [0  0  0  1  0  0]
x(S = 2,right) = [0  0  0  0  1  0]
x(S = 2 , left) = [0  0  0  0  0  1]\
So why is it written that all the states appear identical under the function approximation?
You can choose those states, but is the agent aware of the state it is in? From the text, it seems that the agent cannot distinguish between the three states. Its observation function is completely uninformative.
This is why a stochastic policy is what is needed. This is common for POMDPs, whereas for regular MDPs we can always find a deterministic policy that is guaranteed to be optimal.