Ranking at depth 6
Model's reasoning and response:
To solve this problem, we need to compare each state against the goal state and determine how many constraints are satisfied. The goal state is:

1. Block 'c' is on the table.
2. Block 'a' is on block 'c'.
3. Block 'b' is on block 'a'.
4. Block 'd' is on block 'b'.

Let's evaluate each state:

### State 17
- Block 'c' is not on the table (it's supporting 'a').
- Block 'a' is on block 'c'. (Constraint 2 satisfied)
- Block 'b' is on the table, not on 'a'.
- Block 'd' is on block 'b'. (Constraint 4 satisfied)

**Constraints satisfied: 2, 4**

### State 18
- Block 'c' is on the table. (Constraint 1 satisfied)
- Block 'a' is on the table, not on 'c'.
- Block 'b' is on the table, not on 'a'.
- Block 'd' is on block 'b'. (Constraint 4 satisfied)

**Constraints satisfied: 1, 4**

### State 19
- Block 'c' is on block 'a', not on the table.
- Block 'a' is on the table, not on 'c'.
- Block 'b' is on the table, not on 'a'.
- Block 'd' is on block 'b'. (Constraint 4 satisfied)

**Constraints satisfied: 4**

### State 20
- Block 'c' is on the table. (Constraint 1 satisfied)
- Block 'a' is on the table, not on 'c'.
- Block 'b' is on the table, not on 'a'.
- Block 'd' is on the table, not on 'b'.

**Constraints satisfied: 1**

### State 21
- Block 'c' is on block 'a', not on the table.
- Block 'a' is on the table, not on 'c'.
- Block 'b' is on the table, not on 'a'.
- Block 'd' is on the table, not on 'b'.

**Constraints satisfied: None**

### State 22
- Block 'c' is on the table. (Constraint 1 satisfied)
- Block 'a' is on block 'c'. (Constraint 2 satisfied)
- Block 'b' is on the table, not on 'a'.
- Block 'd' is on the table, not on 'b'.

**Constraints satisfied: 1, 2**

### State 23
- Block 'c' is on the table. (Constraint 1 satisfied)
- Block 'a' is on block 'c'. (Constraint 2 satisfied)
- Block 'b' is on the table, not on 'a'.
- Block 'd' is on the table, not on 'b'.

**Constraints satisfied: 1, 2**

### Ranking
Based on the number of constraints satisfied, the ranking from best to worst is:

```ranking
22, 23, 17, 18, 19, 20, 21
```

States 22 and 23 satisfy two constraints, while the others satisfy fewer.

Final ranking:
Rank 1: State ID 22
State Description: The hand is now empty. Block 'a' is on block 'c' and is clear. Block 'c' is no longer clear. Blocks b and d remain on the table and clear.

Rank 2: State ID 23
State Description: The hand is empty. Block 'a' is on block 'c', making block 'a' not clear. Blocks 'b' and 'd' remain on the table and clear. Block 'c' is no longer clear as it supports block 'a'.

Rank 3: State ID 17
State Description: The hand is empty. Block 'a' is on block 'c' and is clear. Block 'b' is on the table with block 'd' on top of it. Block 'c' is supporting block 'a' and is no longer clear. Block 'd' is on block 'b' and is clear.

Rank 4: State ID 18
State Description: The hand is empty. Block c is on the table and clear. Block a is on the table and clear. Block b is on the table with block d on top of it. Block d is on block b and is clear. The diagram now shows block c on the table, separate from the other blocks.

Rank 5: State ID 19
State Description: The hand is now empty. Block c is stacked on block a, and block c is no longer clear. Block a is no longer clear as it supports block c. Block b remains on the table with block d on top of it. Block d is clear.

Rank 6: State ID 20
State Description: The hand is empty. Blocks a, b, c, and d are on the table and clear. Block c is now on the table and clear, reflecting the change in the diagram.

Rank 7: State ID 21
State Description: The hand is empty. Block 'c' is on block 'a' and is clear. Blocks 'b' and 'd' are still on the table and clear. Block 'a' is no longer clear but is on the table. The diagram now shows 'c' on top of 'a'.

