
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
Positive integers $a$, $b$, $c$, and $d$ are chosen so that $a < b < c < d$, and the system of equations \[
x + 2y = 2023\quad\text{and}\quad y = |x-a| + |x-b| + |x-c| + |x-d|
\] has exactly one solution. What is the minimum value of $d$?

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