
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]


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Assume that the differential equation  $$ y'''+p(x)y''+q(x)y'+r(x)y=0 $$  has solutions  $y_1(x)$ ,  $y_2(x)$ ,  $y_3(x)$  on the real line such that  $$ y_1(x)^2+y_2(x)^2+y_3(x)^2=1 $$  for all real  $x$ . Let  $$ f(x)=y_1'(x)^2+y_2'(x)^2+y_3'(x)^2. $$  Find constants  $A$  and  $B$  such that  $f(x)$  is a solution to the differential equation  $$ y'+Ap(x)y=Br(x). $$  

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