
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
Suppose that  $ P_1(x)\equal{}\frac{d}{dx}(x^2\minus{}1),\ P_2(x)\equal{}\frac{d^2}{dx^2}(x^2\minus{}1)^2,\ P_3(x)\equal{}\frac{d^3}{dx^3}(x^2\minus{}1)^3$ .

Find all possible values for which  $ \int_{\minus{}1}^1 P_k(x)P_l(x)\ dx\ (k\equal{}1,\ 2,\ 3,\ l\equal{}1,\ 2,\ 3)$  can be valued.

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