
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 $E: x^2 = 4y$, and a chord $AB$ of parabola $E$ passes through point $M(1, 4)$ with $M$ as the midpoint,
(1) Find the equation of the line containing chord $AB$.
(2) If line $l: y = x + b$ is tangent to the parabola $E$ at point $P$, find the equation of the circle with center $P$ that is tangent to the directrix of parabola $E$.

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