
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
Let  $a,b$  be constant numbers such that  $0<a<b.$  If a function  $f(x)$  always satisfies  $f'(x) >0$  at  $a<x<b,$  for  $a<t<b$  find the value of  $t$  for which the following the integral is minimized.

\[ \int_a^b |f(x)-f(t)|x\ dx. \]

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