
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  $a_n$  be a sequence defined by some  $a_0$  and the recursion  $a_{n+1} = a_n + 2 \cdot  3^n$  for  $n \ge 0$ . 
Determine all rational values of  $a_0$  such that  $a^j_k / a^k_j$  is an integer for all integers  $j$  and  $k$  with  $0 < j < k$ .

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