
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
Given the set $A=\{x|x^2-3x+2\leq0\}$, and the function $f(x)=x^2-2ax+1$.
1. Solve the inequality $f(x)\leq3a^2+1$ when $a\neq0$;
2. If the proposition "There exists $x_0 \in A$ such that $f(x_0) \leq A$" is false, find the range of the real number $a$.

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