
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
Define: If the point $P(m,n)$ satisfies $2m-n=1$, then point $P$ is called a coordinate point of the linear equation $2x-y=1$.  
$(1)$ If the point $A(3,a)$ is a coordinate point of the equation $2x-y=1$, then $a=$______.  
$(2)$ If $B(b+c,b+5)$ is a coordinate point of the equation $2x-y=1$, and $b$, $c$ are positive integers, find the values of $b$ and $c$.

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