
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
Let  $p> 3$  be a prime number. Determine the number of all ordered sixes  $(a, b, c, d, e, f)$  of positive integers whose sum is  $3p$  and all fractions  $\frac{a + b}{c + d},\frac{b + c}{d + e},\frac{c + d}{e + f},\frac{d + e}{f + a},\frac{e + f}{a + b}$  have integer values.

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