
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 circle $C_{1}$ has its center at the origin and is tangent to the line $3x+4y-10=0$, and the line $l$ passes through the point $M(1,2)$.  
$(1)$ Find the standard equation of circle $C_{1}$.  
$(2)$ If the chord of circle $C_{1}$ intercepted by line $l$ has a length of $2\sqrt{3}$, find the equation of line $l$.

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