
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 $AB$ of fixed length $m$ whose endpoints are moving on the right branch of the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ $(m>\frac{2 b^{2}}{a})$, find the minimum value of the x-coordinate of the midpoint $M$ of $AB$ in terms of $a, b, m$.

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