
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 that the moving circle $P$ is internally tangent to the fixed circle $B$: $x^{2}+y^{2}+2 \sqrt {5}x-31=0$, and the moving circle $P$ passes through a certain point $A( \sqrt {5},0)$.
$(1)$ Find the equation of the trajectory $E$ of the moving circle's center $P$;
$(2)$ If a point $(x,y)$ is on the trajectory $E$, find the range of values for $x+2y$.

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