
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 a triangle $\triangle ABC$ with angles $A$, $B$, $C$ and their respective opposite sides $a$, $b$, $c$ that satisfy the equation $\frac{\sqrt{2}a - b}{c} = \frac{\cos B}{\cos C}$.

1. Find the measure of angle $C$.
2. Given the function $f(x) = \cos(2x + C)$, the graph of $f(x)$ is translated to the right by $\frac{\pi}{4}$ units to obtain the graph of function $g(x)$. Find the range of $g(x)$ on the interval $\left[0, \frac{\pi}{3}\right]$.

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