
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 $ \dfrac {1}{3}\leqslant a\leqslant 1$, if the function $f(x)=ax^{2}-2x+1$ is defined on the domain $[1,3]$.  
$(1)$ Find the minimum value of $f(x)$ on its domain (expressed in terms of $a$);  
$(2)$ Let the maximum value of $f(x)$ on its domain be $M(a)$, and the minimum value be $N(a)$, find the minimum value of $M(a)-N(a)$.

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