
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 internal angles $A$, $B$, and $C$ opposite sides $a$, $b$, and $c$, respectively. Given that $a=1$, $b=2$, and $\cos C= \frac{1}{4}$.

$(1)$ Find the perimeter of $\triangle ABC$;

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

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