
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
In a certain circle, the chord of a  $d$ -degree arc is 22 centimeters long, and the chord of a  $2d$ -degree arc is 20 centimeters longer than the chord of a  $3d$ -degree arc, where  $d<120.$   The length of the chord of a  $3d$ -degree arc is  $-m+\sqrt{n}$  centimeters, where  $m$  and  $n$  are positive integers.  Find  $m+n.$ 

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