
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 a ray passes through the intersection point $M$ of the known lines $l_{1}:3x-y+7=0$ and $l_{2}:2x+y+3=0$, and after hitting a point $N(1,0)$ on the $x$-axis, it is reflected by the $x$-axis.

(1) Find the coordinates of the point $P$, which is the symmetric point of $M$ with respect to the $x$-axis.

(2) Find the equation of the line $l_{3}$ where the reflected ray lies.

(3) Find the equation of a line that is at a distance of $\sqrt{10}$ from $l_{3}$.

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