
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 two circles $C\_1: (x+1)^2 + y = 25$ and $C\_2: (x-1)^2 + y = 1$, a moving circle $C$ is tangent internally to both $C\_1$ and $C\_2$.

1. Find the equation of the trajectory $E$ of the center of the moving circle $C$.
2. Point $P(1, t)$ lies on the trajectory $E$ and is in the first quadrant. Two lines are drawn through point $P$ intersecting the trajectory $E$ at points $A$ and $B$. If the slopes of lines $PA$ and $PB$ are negative reciprocals of each other, is the slope of line $AB$ a constant value? If so, find the constant value; if not, explain the reason.

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