
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 that $a > 0$, the endpoints of the graph of the function $f(x)=x-\frac{a}{x}\ (x \in [1,2])$ are $A$ and $B$. Let $M$ be any point on the graph of $f(x)$, draw a line $l$ perpendicular to the $x$-axis passing through $M$, and let $N$ be the point of intersection of $l$ and the line segment $AB$. If $|MN| \leqslant 1$ always holds true, then the maximum value of $a$ is \_\_\_\_\_\_.

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