
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
**6.** Let  $f(x)$  be an arbitrary function, differentiable infinitely many times. Then the  $n$ th derivative of  $f(e^{x})$  has the form $\frac{d^{n}}{dx^{n}}f(e^{x})= \sum_{k=0}^{n} a_{kn}e^{kx}f^{(k)}(e^{x})$  ( $n=0,1,2,\dots$ ).

From the coefficients  $a_{kn}$  compose the sequence of polynomials $P_{n}(x)= \sum_{k=0}^{n} a_{kn}x^{k}$  ( $n=0,1,2,\dots$ )

and find a closed form for the function $F(t,x) = \sum_{n=0}^{\infty} \frac{P_{n}(x)}{n!}t^{n}.$ **(S. 22)**

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