
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 a circle of circumference $12$ units, a tangent and a secant are drawn from an external point $P$. The secant divides the circle into unequal arcs with lengths $m$ and $n$. It is given that the length of the tangent, $t$, is given by $2\sqrt{mn}$, $m > 2n$, and both $m$ and $t$ are integers. Determine the possible number of values for $t$.

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