
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
To promote the Hangzhou Asian Games in 2023, a public service advertising company used a 1-meter long iron wire to make a right-angled triangle poster with the ends connected. Find:<br/>$(1)$ What is the shortest length of the hypotenuse of the poster?<br/>$(2)$ If an inscribed circle is drawn on the poster, what is the maximum value of the radius $r$ of the inscribed circle in the right-angled triangle?

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