
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
A line $l$ passes through the intersection point of the lines $3x+y-1=0$ and $x-5y-11=0$, and is perpendicular to the line $x+4y=0$.  
(1) Find the equation of line $l$;  
(2) Find the length of the chord $|AB|$ cut by the circle $x^2+(y-11)^2=25$ on line $l$.

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