
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 the polynomials  $P(x)=px^4+qx^3+rx^2+sx+t,\ Q(x)=\frac{d}{dx}P(x)$ , find the real numbers  $p,\ q,\ r,\ s,\ t$  such that  $P(\sqrt{-5})=0,\ Q(\sqrt{-2})=0$  and  $\int_0^1 P(x)dx=-\frac{52}{5}.$ 

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