
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 $M$ is $\left(a,0\right)$ with $\left(a\leqslant 0\right)$, it passes through point $P\left(0,-2\right)$, and is tangent to the line $x+y+2\sqrt{2}=0$.  
$(1)$ Find the standard equation of circle $M$.  
$(2)$ If a line $l$ passing through point $Q\left(0,1\right)$ with slope $k$ intersects circle $M$ at points $A$ and $B$, and the length of chord $AB$ is $\sqrt{14}$, find the equation of line $l$.

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