
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
In the Cartesian coordinate system, $A$ and $B$ are two points on the line $x+y=m$, and $|AB|=10$. If for any point $P\left(\cos \theta ,\sin \theta \right)\left(0\leqslant \theta\ \ \lt 2\pi \right)$, there exist $A$ and $B$ such that $\angle APB=90^{\circ}$ holds, find the maximum value of $m$.

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