
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 $M$ be a four-digit number, where the digit in the thousands place is $a$, the digit in the hundreds place is $b$, the digit in the tens place is $1$, and the digit in the ones place is $c$ ($c\neq 0$). If the point $(1,c)$ lies on the line $y=ax+b$, then the line $y=ax+b$ is called the interactive line of $M$. Moving the digit $c$ of $M$ to the front of the thousands place digit $a$ creates a new four-digit number $N$. For example, if $M=2315$, then $N=5231$. Let $F(M)=\frac{M+N}{11}$. Find $F(2517)=\_\_\_\_\_\_$. If $F(M)$ is a multiple of $6$, then the number of interactive lines of $M$ that satisfy the condition is ______.

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