
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 $S$ be the set of seven-element ordered arrays $\left(a_{1}, a_{2}, \ldots, a_{7}\right)$, where $a_i = 1$ or 0. For any two elements $a = \left(a_{1}, a_{2}, \ldots, a_{7}\right)$ and $b = \left(b_{1}, b_{2}, \ldots, b_{7}\right)$ in $S$, define the distance between $a$ and $b$ as $\sum_{i=1}^{7}\left|a_{i}-b_{i}\right|$. Let $T$ be a subset of $S$ such that the distance between any two elements is at least 3. Find the maximum value of $|T|$.

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