
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 the parabola $C_{1}$: $y^{2}=16x$ and the circle $C_{2}$: $\left(x-4\right)^{2}+y^{2}=1$, where point $M$ has coordinates $\left(8,0\right)$, and points $P$ and $Q$ are moving points on $C_{1}$ and $C_{2}$ respectively, satisfying $|PM|=|PQ|$. Find the range of the x-coordinate of point $P$.

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