
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
Given points $A(1,3)$, $B(4,2)$, and $C(1,-7)$, let $M$ be the circle passing through these points, intersecting the $y$-axis at points $P$ and $Q$.
1. Find the length of the line segment $PQ$.
2. Circle $N$ has its center moving along the line $2x - y + 6 = 0$ with a radius of $10$. If circle $N$ intersects circle $M$, find the range of possible values for the $x$-coordinate $a$ of the center of circle $N$.

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