
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
Find the set of all real values of  $a$  for which the real polynomial equation  $P(x)=x^2-2ax+b=0$  has real roots, given that  $P(0)\cdot P(1)\cdot P(2)\neq 0$  and  $P(0),P(1),P(2)$  form a geometric progression.

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