
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 function $f(x)=ax^{2}-2x+1$.
$(1)$ When $a\neq 0$, discuss the monotonicity of the function $f(x)$;
$(2)$ If $\frac {1}{3}\leqslant a\leqslant 1$, and the maximum value of $f(x)$ on $[1,3]$ is $M(a)$, the minimum value is $N(a)$, let $g(a)=M(a)-N(a)$, find the expression of $g(a)$;
$(3)$ Under the condition of $(2)$, find the minimum value of $g(a)$.

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