
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 $a_{i}\neq 0$ for $i=1,2,\ldots ,2024$ and $\frac{|{a}_{1}|}{{a}_{1}}+\frac{|{a}_{2}|}{{a}_{2}}+\frac{|{a}_{3}|}{{a}_{3}}+\cdots+\frac{|{a}_{2023}|}{{a}_{2023}}+\frac{|{a}_{2024}|}{{a}_{2024}}=2000$, then the probability that the graph of the linear function $y=a_{i}x+i$ passes through the first, second, and fourth quadrants for $a_{i}$ is ______.

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