
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 the hyperbola $\frac{x^{2}}{4} - \frac{y^{2}}{5} = 1$, let $F\_1$ be its left focus. A circle with equation $x^{2} + y^{2} = 4$ intersects the right branch of the hyperbola at point $P$ and has a tangent point $T$. If $M$ is the midpoint of $PF\_1$, find $|MO| - |MT|$ where $O$ is the origin.

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