
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 line segment \( AB \) with a fixed length \( m \), both endpoints of which move along the right branch of the hyperbola \( \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1 \) (with \( m > \frac{2 b^{2}}{a} \)), find the minimum x-coordinate of the midpoint \( M \) of \( AB \) expressed in terms of \( a \), \( b \), and \( m \).

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