
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
Define three sequences  $a_n,b_n,c_n$  as  $a_0=b_0=c_0=1$  and
\begin{align*}
    a_{n+1}&=a_n+3b_n+3c_n 
    b_{n+1}&=a_n+b_n+3c_n 
    c_{n+1}&=a_n+b_n+c_n
\end{align*}
for  $n\geq0$ . Let  $A,B,C$  be the remainders when  $a_{13^4},b_{13^4},c_{13^4}$  are divided by  $13$ . Find the ordered triple  $\left(A,B,C\right)$ .

*2019 CCA Math Bonanza Team Round #10*

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