
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 tangent to the circle \(x^2 + y^2 = 9\) is drawn from the left focus \(F\) of the hyperbola \(\frac{x^2}{9} - \frac{y^2}{16} = 1\), with the point of tangency being \(T\). The line segment \(FT\) is extended to intersect the right branch of the hyperbola at point \(P\). Let \(M\) be the midpoint of the line segment \(FP\) and \(O\) be the origin. Find the value of \(|MO - MT|\).

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