
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  $c$  be a real number. If the inequality  $$ f(c)\cdot f(-c)\ge f(a) $$  holds for all  $f(x)=x^2-2ax+b$  where  $a$  and  $b$  are arbitrary real numbers, find all possible values of  $c$ .

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