
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
Let the circumference of the circle be 10. From a point $P$ outside the circle, a tangent to the circle is drawn. Its length is denoted by $t$. A secant from $P$ divides the circle into two unequal arcs with lengths $m$ and $n$. It is known that $t$ is the geometric mean of $m$ and $n$. If both $m$ and $t$ are integers, calculate the number of possible values that $t$ can have.

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