
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 the solution set of the inequality $ax^2+2|x-a|-20<0$ with respect to $x$ be $A$. Investigate whether there exists a natural number $a$ such that the solutions of the inequalities $x^2+x-2<0$ and $|2x-1|<x+2$ both belong to $A$. If no such $a$ exists, explain why. If such $a$ exists, find all possible values of $a$.

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