
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
For how many primes $p$ less than  $15$ , determine the number of integer triples $(m,n,k)$ such that 
\[
\begin{array}{rcl}
m+n+k &\equiv& 0 \pmod p 
mn+mk+nk &\equiv& 1 \pmod p 
mnk &\equiv& 2 \pmod p.
\end{array}

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