
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 $$\frac {1}{3} \leq a \leq 1$$, if the function $f(x) = ax^2 - 2x + 1$ has a domain of $[1, 3]$.  
(1) Find the minimum value of $f(x)$ in its domain (expressed in terms of $a$);  
(2) Let the maximum value of $f(x)$ in 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