
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
 $(a)$  Compute -
\begin{align*}
\frac{\mathrm{d}}{\mathrm{d}x} \bigg[ \int_{0}^{e^x} \log ( t ) \cos^4 ( t ) \mathrm{d}t \bigg]
\end{align*} $(b)$  For  $x > 0 $  define  $F ( x ) = \int_{1}^{x} t \log ( t ) \mathrm{d}t . $  $1.$  Determine the open interval(s) (if any) where  $F ( x )$  is decreasing and all the open interval(s) (if any) where  $F ( x )$  is increasing. $2.$  Determine all the local minima of  $F ( x )$  (if any) and all the local maxima of  $F ( x )$  (if any)  $.$ 

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