
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 the circle $C: x^{2}+y^{2}=25$ and point $P(3,4)$.  
$(1)$ Find the equation of the tangent line $l$ passing through point $P$ on circle $C$.  
$(2)$ If the line $m$ passes through point $P$ and intersects circle $C$ with a chord length of $8$, find the equation of line $m$.

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