
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
Given the sequence $\{a_{n}\}$ where ${a}_{1}=\frac{1}{2}$, ${a}_{4}=\frac{1}{8}$, and $a_{n+1}a_{n}+a_{n-1}a_{n}=2a_{n+1}a_{n-1}$ for $n\geqslant 2$, find the value of $T_{2024}$, where $T_{n}$ is the sum of the first $n$ terms of the sequence $\{b_{n}\}$ and $b_{n}=a_{n}a_{n+1}$.

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