
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
How many  $6$ -tuples  $(a, b, c, d, e, f)$  of natural numbers are there for which  $a>b>c>d>e>f$  and  $a+f=b+e=c+d=30$  are simultaneously true?

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