
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 lines $l_{1}: x-y+1=0$ and $l_{2}: (a-1)x+ay+\frac{1}{2}=0$.
1. If $l_{1} \parallel l_{2}$, find the value of the real number $a$.
2. Under the condition of (1), suppose $l_{1}$ and $l_{2}$ intersect the x-axis at points $A$ and $B$ respectively, and a moving point $P$ in the plane has the ratio of distances to points $A$ and $B$ as $\sqrt{2}$. Find the equation of the trajectory $E$ of point $P$.

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