
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
(1) Given points $A(-2,0)$ and $B(0,3)$, find a point $P$ on the line $l: x-y+1=0$ such that the value of $|PA|+|PB|$ is minimized. The minimum value is __________.

(2) Given a circle with center at $C(2,-1)$, and a chord intercepted by the line $l: x-y-1=0$ with length $2\sqrt{2}$, the equation of this circle is __________.

(3) Given that $A$ and $B$ are both obtuse angles, and $\sin A= \frac{\sqrt{5}}{5}, \sin B= \frac{\sqrt{10}}{10}$, find the value of $A+B$ as __________.

(4) Given that $|\vec{a}|=|\vec{b}|=2$ and the angle between $\vec{a}$ and $\vec{b}$ is $60^{\circ}$, the projection of $\vec{a} + \vec{b}$ in the direction of $\vec{a}$ is __________.

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