
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 $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ in $\triangle ABC$, respectively, and it satisfies $(2b-a) \cdot \cos C = c \cdot \cos A$.
$(1)$ Find the size of angle $C$;
$(2)$ Let $y = -4\sqrt{3}\sin^2\frac{A}{2} + 2\sin(C-B)$, find the maximum value of $y$ and determine the shape of $\triangle ABC$ when $y$ reaches its maximum value.

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