
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 point  $(a,b)$  with  $0<b<a$ , determine the minimum perimeter of a triangle with one vertex at  $(a,b)$ , one on the  $x$ -axis, and one on the line  $y=x$ .  You may assume that a triangle of minimum perimeter exists.

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