
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
The graph of $y=ax+10$ intersects the x-axis when $y=0$, so $0=ax+10$. This is a linear equation in the variable $x$ for a given value of $a$. Similarly, the graph of $y=4x+b$ intersects the x-axis when $y=0$, so $0=4x+b$. This is also a linear equation in the variable $x$ for a given value of $b$. Since the two graphs intersect at the same point, we must have the same value of $x$ in both equations. Therefore, we must have $a(x)+10=4(x)+b$ for some value of $x$. Since $a$ and $b$ are integers, this equation must be true for all integer values of $x$. We must have $\frac{a-4}{1}x=-b+10$, which is true for all values of $x$ if and only if $a-4=0$ and $-b+10=0$, or $a-4=0$ and $-b+10=-1$, or $a-4=-1$ and $-b+10=0$, or $a-4=-1$ and $-b+10=-1$. Solving these systems of equations, we get $(a,b)=(4,-10), (4,-11), (1,10), (1,9)$. Since $a$ and $b$ are positive integers, we ignore the solutions $(a,b)=(4,-10)$ and $(4,-11)$, so the only possible solutions are $a=1$ and $b=9$ or $a=1$ and $b=10$. If $a=1$ and $b=9$, the graphs of the two lines intersect the x-axis at the point $(x,y)=(x,0)$ where $x$ satisfies the equation $0=x+10$ and $0=4x+9$ or $x=-10$ and $4x=-9$. Simplifying, we get $x=-10$ and $x=-\frac{9}{4}$. Since $x$ must be an integer, the graphs of the lines intersect the x-axis at the point $(-10,0)$. If $a=1$ and $b=10$, the graphs of the lines intersect the x-axis at the point $(x,y)=(x,0)$ where $x$ satisfies the equation $0=x+10$ and $0=4x+10$ or $x=-10$ and $4x=-10$. Simplifying, we get $x=-10$ and $x=-2.5$. Since $x$ must be an integer, the graphs of the lines intersect the x-axis at the point $(-10,0)$. 

Given that the graphs of $y=ax+10$ and $y=4x+b$ intersect the x-axis at the same point, determine the sum of all possible x-coordinates of these points of intersection.

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