
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 $S_{n}$ be the sum of the first $n$ terms of the sequence $\{a_{n}\}$, given that $a_{1}=1$, $\frac{S_n}{a_{n+1}} - \frac{S_n}{a_n} = -\frac{1}{2}$.  
$(1)$ Find the general formula for $\{a_{n}\}$;  
$(2)$ Let ${b_n} = 2^{a_n}$, and denote the sum of the first $n$ terms of the sequence $\{b_{n}\}$ as $T_{n}$. Find the remainder when $T_{2n-1}$ is divided by $3$.

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