
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
On rectangular coordinates, point  $A = (1,2)$ ,  $B = (3,4)$ .  $P = (a, 0)$  is on  $x$ -axis. Given that  $P$  is chosen such that  $AP + PB$  is minimized, compute  $60a$ .

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