
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
The phase plane \(O p q\) is divided by the parabola \(p^{2}-4q=0\) and the lines \(p+q+1=0\), \(-2p+q+4=0\) into several regions. For the points in each region, determine how many roots the corresponding polynomial \(x^{2}+px+q=0\) has in the interval \((-2,1)\).

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