
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
The irrational number  $\alpha>1$  satisfies  $\alpha^2-3\alpha-1=0$ . Given that there is a fraction  $\frac{m}{n}$  such that  $n<500$  and  $\left|\alpha-\frac{m}{n}\right|<3\cdot10^{-6}$ , find  $m$ .

*2018 CCA Math Bonanza Team Round #10*

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