<system>
You are an experienced Python developer. You will be provided with an incorrect code snippet from a Python program. The task this program is supposed to perform is described in the following user prompt.
Your task is to rewrite the program so that it performs the task as expected without any errors. You will be rewarded based on the number of test cases your code passes.
</system>

<user>
## Description

The inverted pendulum swingup problem is based on the classic problem in control theory.
The system consists of a pendulum attached at one end to a fixed point, and the other end being free.
The pendulum starts in a random position and the goal is to apply torque on the free end to swing it
into an upright position, with its center of gravity right above the fixed point.

-  `x-y`: cartesian coordinates of the pendulum's end in meters.
- `theta` : angle in radians.
- `tau`: torque in `N m`. Defined as positive _counter-clockwise_.

## Action Space

The action is a `ndarray` with shape `(1,)` representing the torque applied to free end of the pendulum.

| Num | Action | Min  | Max |
|-----|--------|------|-----|
| 0   | Torque | -2.0 | 2.0 |


## Observation Space

The observation is a `ndarray` with shape `(3,)` representing the x-y coordinates of the pendulum's free
end and its angular velocity.

| Num | Observation      | Min  | Max |
|-----|------------------|------|-----|
| 0   | x = cos(theta)   | -1.0 | 1.0 |
| 1   | y = sin(theta)   | -1.0 | 1.0 |
| 2   | Angular Velocity | -8.0 | 8.0 |

## Rewards

The reward function is defined as:

*r = -(theta<sup>2</sup> + 0.1 * theta_dt<sup>2</sup> + 0.001 * torque<sup>2</sup>)*

where `theta` is the pendulum's angle normalized between *[-pi, pi]* (with 0 being in the upright position).
Based on the above equation, the minimum reward that can be obtained is
*-(pi<sup>2</sup> + 0.1 * 8<sup>2</sup> + 0.001 * 2<sup>2</sup>) = -16.2736044*,
while the maximum reward is zero (pendulum is upright with zero velocity and no torque applied).

## Starting State

The starting state is a random angle in *[-pi, pi]* and a random angular velocity in *[-1,1]*.

## Class Definition
The class should be called "Environment". It should have at least:

- an __init__ function to set up the Environment, which defines all the variables described in the above documentation, plus any additional variables needed to maintain the environment state or to implement its functionality.
- a set_state function to set a custom value for the environment and its internal representation (you can assume that when "set_state" is used, the task is not done and internal variables should be set as a consequence). set_state takes a single argument as input: a state observation from the observation space defined above.
- a step function to predict a step in the environment. The input parameters for the step function are:
    - An action, which must be contained in the action space described above.
  
    The outputs required by the step function are:
    - An observation, which must be contained in the observation space described above.
    - The reward for taking the action, as described in the reward definition above.
    - A boolean variable indicating if the episode is done.

## Important Notes
Only produce the environment class, containing the __init__, set_state and step functions and any additional functions you may need to complete this task. Do not write an example of how to use the class or anything else.
Be careful about edge cases.
Make sure to write all the required functions and that they have the exact names as specified in the task description. Missing or incorrectly named functions will not pass the tests and will result in a score of 0.
It is of VITAL importance that you do not leave undefined any function, but implement each of them completely.

First, write an explanation of the difference between the ground-truth transition and the step function's outputs in the example provided.
Second, point out the part of the code responsible for the incorrect prediction and why its logic is erroneous.
Third, suggest a concrete, actionable fix for it. 
Finally, fix the program in its entirety following the suggestion. The expected output is in the format:
## Error explanation
[your explanation of the error]
    
## Error location and wrong logic
[where the error comes from and why]
    
## Fix suggestion
[how to fix the error]
    
## Correct code
[your code]

## Incorrect code
You are provided with the following code snippet to fix.
```python
{CODE}
```
The code's step function additionally makes a wrong prediction about this transition.
## Step function inputs
{TRANSITION_INPUTS}
    
## Ground-truth transition
{GT_PREDICTION}
    
## Step function's incorrect outputs
{PREDICTION}
</user>