
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
Let $\triangle ABC$ have sides $a$, $b$, $c$ opposite to angles $A$, $B$, $C$ respectively. Given that $\frac{{a^2 - b^2}}{{c^2}} = \frac{{a^2 + b^2 - c^2}}{{ab}}$. Find:<br/>
$(1)$ If $C = \frac{\pi}{4}$, determine the values of angles $A$ and $B$.<br/>
$(2)$ If $\triangle ABC$ is an acute triangle, find the range of values for $\frac{a}{{b\cos^2B}}$.

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