
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
If an infinite sequence $\{a_{n}\}$ satisfies: whenever $a_{p}=a_{q}(p,q∈{N}^{*})$, then $a_{p+k}=a_{q+k}$, the sequence $\{a_{n}\}$ is called a "$k$-order equivalent progressive sequence". If the sequence $\{a_{n}\}$ is a "$1$-order equivalent progressive sequence", and $a_{1}=1$, $a_{3}=3$, $a_{5}=1$, $a_{4}+a_{7}+a_{10}=2$, then $a_{2023}=$______, let ${b}_{n}=(-1)^{n}{a}_{n}$, the sum of the first $n$ terms of the sequence $\{b_{n}\}$ is $S_{n}$, then $S_{2024}=\_\_\_\_\_\_$.

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