
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
A quadruplet of distinct positive integers  $(a, b, c, d)$  is called  $k$ -good if the following conditions hold:

1. Among  $a, b, c, d$ , no three form an arithmetic progression.

2. Among  $a+b, a+c, a+d, b+c, b+d, c+d$ , there are  $k$  of them, forming an arithmetic progression. $a)$  Find a  $4$ -good quadruplet. $b)$  What is the maximal  $k$ , such that there is a  $k$ -good quadruplet?

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