
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Consider the two points \( A(4,1) \) and \( B(2,5) \). For each point \( C \) with positive integer coordinates, we define \( d_C \) to be the shortest distance needed to travel from \( A \) to \( C \) to \( B \) moving only horizontally and/or vertically. For example, for the point \( C(3,4) \), we compute \( d_C \) as follows:

- To get from \( A \) to \( C \) moving only horizontally and/or vertically, we can move 1 unit to the left then 3 units up for a total distance of \( 1+3=4 \).
- The shortest path from \( C \) to \( B \) moving only horizontally and/or vertically is to move 1 unit left and then 1 unit up (or 1 unit up then 1 unit left) for a total distance of \( 1+1=2 \).

Thus, for \( C(3,4) \), we have that \( d_C = 4 + 2 = 6 \).

The positive integer \( N \) has the property that there are exactly 2023 points \( C(x, y) \) with \( x > 0 \) and \( y > 0 \) and \( d_C = N \). What is the value of \( N \)?

Present the answer in LaTex format: \boxed{Your answer}
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