
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 quadruple  $(a,b,c,d)$   of distinct integers is said to be  $balanced$  if  $a+c=b+d$ . Let  $\mathcal{S}$   be any set of quadruples  $(a,b,c,d)$  where  $1 \leqslant a<b<d<c \leqslant 20$  and where the cardinality of  $\mathcal{S}$   is  $4411$ .  Find the least number of balanced quadruples in  $\mathcal{S}.$ 

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