
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 equation of the line $l$ is $\left(a+1\right)x+y-5-2a=0\left(a\in R\right)$.  
$(1)$ Find the coordinates of the fixed point $P$ that the line $l$ passes through.  
$(2)$ The line $l$ intersects the positive half of the $x$-axis and the positive half of the $y$-axis at points $A$ and $B$, respectively. When the area of $\triangle AOB$ is minimized, find the equation of the line $l$.

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