
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 $a$ and $b$ be positive integers. If the graph of the quadratic function $y=a^2+bx+1$ intersects the x-axis at two points, and the x-coordinates of these two points, $x_1$ and $x_2$, satisfy $-1<x_1<x_2<0$, find the minimum values of the positive integers $a$ and $b$, and the values of $x_1$ and $x_2$ at this time.

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