
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
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. It is known that vectors $\vec{m}=(\cos A-2\cos C,2c-a)$ and $\vec{n}=(\cos B,b)$ are parallel.

$(1)$ Find the value of $\frac{\sin C}{\sin A}$;

$(2)$ If $b\cos C+c\cos B=1$ and the perimeter of $\triangle ABC$ is $5$, find the length of $b$.

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