
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 \( \frac{1}{3} \leqslant a \leqslant 1 \), if \( f(x)=a x^{2}-2 x+1 \) attains its maximum value \( M(a) \) and minimum value \( N(a) \) on the interval \([1,3]\), and let \( g(a)=M(a)-N(a) \), then the minimum value of \( g(a) \) is \(\quad\) .

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