
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 the set
$$
\begin{aligned}
A= & \left\{\left(a_{1}, a_{2}, \cdots, a_{8}\right) \mid a_{i} \in \mathbf{Z}_{+}, 1 \leqslant a_{i} \leqslant i+1 ; i=1,2, \cdots, 8\} .
\end{aligned}
$$

Consider a subset $X \subset A$. If each pair of distinct elements $\left(a_{1}, a_{2}, \cdots, a_{8}\right)$ and $\left(b_{1}, b_{2}, \cdots, b_{8}\right)$ in $X$ differ in at least three positions (i.e., there exist at least three indices $i$ such that $a_{i} \neq b_{i}$), then $X$ is called a "good set". Find the maximum possible size of a good subset of $A$.

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