
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
In a Cartesian coordinate system, it is known that point $A$ is at $(1, 0)$ and point $B$ moves along the line $l: x = -1$. The line perpendicular to $l$ through point $B$ intersects the perpendicular bisector of segment $AB$ at point $M$.  
(1) Find the equation for the locus $E$ of the moving point $M$;  
(2) Construct the tangent to the locus $E$ at point $P(1, 2)$ and find the equation of the tangent line.

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