
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
Find arithmetic sequences of natural numbers 

$$
\begin{array}{llllll}
a_{1}, & a_{2}, & a_{3}, & \ldots, & a_{k}, & \ldots \\
b_{1}, & b_{2}, & b_{3}, & \ldots, & b_{k}, & \ldots \\
c_{1}, & c_{2}, & c_{3}, & \ldots, & c_{k}, & \ldots \\
d_{1}, & d_{2}, & d_{3}, & \ldots, & d_{k}, & \ldots
\end{array}
$$

such that in the sequence formed by their union

$$
a_{1}, b_{1}, c_{1}, d_{1}, a_{2}, b_{2}, c_{2}, d_{2}, \ldots, a_{k}, b_{k}, c_{k}, d_{k}, \ldots
$$

any element after $a_{1}$ is the sum of the immediately preceding element and the last digit of that preceding element.

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