
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 planar region defined by the system of inequalities:
\begin{cases} x \geqslant 0 \\ y \geqslant 0 \\ x + 2y - 4 \leqslant 0 \end{cases}
which is exactly covered by the smallest circle $(C)$: $(x - a)^2 + (y - b)^2 = r^2$ and its interior.

1. Find the equation of circle $(C)$.
2. If a line $(l)$ with slope $1$ intersects circle $(C)$ at two distinct points $A$ and $B$ such that $CA \perp CB$, find the equation of line $(l)$.

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