
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
Suppose in triangle $ABC$, the sides opposite the internal angles $A$, $B$, $C$ are $a$, $b$, $c$ respectively. Given $a=1$, $b=2$, $\cos C = \frac{1}{2}$.

(1) Find the perimeter of triangle $ABC$.

(2) Find the value of $\cos(A - C)$.

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