
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  $P(x), Q(x), $  and  $R(x)$  be three monic quadratic polynomials with only real roots, satisfying   $$ P(Q(x))=(x-1)(x-3)(x-5)(x-7) $$    $$ Q(R(x))=(x-2)(x-4)(x-6)(x-8) $$      for all real numbers  $x.$  What is  $P(0)+Q(0)+R(0)?$ *Proposed by Kyle Lee*

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