
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
From the left focus $F$ of the hyperbola $\frac{x^{2}}{3}-\frac{y^{2}}{5}=1$, a tangent line $FP$ is drawn to the circle $x^{2}+y^{2}=3$. Let $P$ be the point of intersection of the tangent line $FP$ and the right branch of the hyperbola, $T$ be the point of tangency, $M$ be the midpoint of the line segment $FP$, and $O$ be the origin. Find $|MO|-|MT|$.

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