
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
The sequence of positive integers ${a_n}$ satisfies ${a_{n+1}} = \begin{cases} \frac{1}{2}a_n, & \text{if $a_n$ is even} \\ 3a_n + 1, & \text{if $a_n$ is odd} \end{cases}$, and it is known that $a_7 = 2$. Let $S$ be the maximum sum of the first 7 terms of the sequence ${a_n}$, and let $T$ be the sum of all possible values of $a_1$ arranged in increasing order to form a new sequence ${b_n}$. Find the value of $S - T$.

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