
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 \(d\) (where \(a<b<c<d\)) are chosen such that the system of equations 
\[ 
3x + y = 3005\quad\text{and}\quad y = |x-a| + |x-b| + |x-c| + |x-d| 
\]
has exactly two solutions. Determine the minimum value of \(d\).

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