
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
Shelly writes down a vector \( v = (a, b, c, d) \), where \( 0 < a < b < c < d \) are integers. Let \( \sigma(v) \) denote the set of 24 vectors whose coordinates are \( a, b, c \), and \( d \) in some order. For instance, \( \sigma(v) \) contains \( (b, c, d, a) \). Shelly notes that there are 3 vectors in \( \sigma(v) \) whose sum is of the form \( (s, s, s, s) \) for some \( s \). What is the smallest possible value of \( d \) ?

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