
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 center of circle $M$ lies on the $y$-axis, the radius is $1$, and the chord intercepted by line $l: y = 2x + 2$ on circle $M$ has a length of $\frac{4\sqrt{5}}{5}$. Additionally, the circle center $M$ is located below line $l$.

(1) Find the equation of circle $M$;

(2) Let $A(t, 0), B(t + 5, 0) \, (-4 \leqslant t \leqslant -1)$, if $AC, BC$ are tangent lines to circle $M$, find the minimum value of the area of $\triangle ABC$.

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