
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 that $\frac{1}{3} \leq a \leq 1$, if the function $f(x) = ax^2 - 2x + 1$ has a maximum value of $M(a)$ and a minimum value of $N(a)$ in the interval $[1, 3]$, let $g(a) = M(a) - N(a)$.
(1) Find the analytic expression for $g(a)$;
(2) Write down the monotonic interval of function $g(a)$ without proof and find the maximum value of $g(a)$.

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