
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
Let \( A = \{(x, y) \mid x, y \in \mathbf{N}, 1 \leq x, y \leq 20\} \) and \( B = \{(x, y) \mid x, y \in \mathbf{N}, 2 \leq x, y \leq 19\} \). Points in \( A \) are colored either red or blue, with a total of 219 red points, 180 of which are within \( B \). The four corner points \((1,1)\), \((1,20)\), \((20,1)\), and \((20,20)\) are all colored blue. Adjacent grid points are connected with red, blue, or black line segments according to the following rules: connect with a red line if both points are red, with a blue line if both points are blue, and with a black line if one point is red and the other is blue. If the number of black line segments of length 1 is 237, how many blue line segments of length 1 are there?

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