
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
Read the following materials and solve the related problems.

We know that the expansion of $(a+b)^{2}$ is $a^{2}+2ab+b^{2}$. We can use the polynomial multiplication rule to expand $(a+b)^{3}$. If we go further and want to expand $(a+b)^{4}$, $(a+b)^{5}$, you will definitely find that solving the above problems requires a lot of calculations. Is there a simpler method? Let's look for a pattern!

If we arrange each term of $(a+b)^{n}$ (where n is a non-negative integer) in descending order according to the number of times the letter $a$ appears, we can get the following equation:

Number of terms in the calculation result and coefficients of each term:

$(a+b)^{0}=1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1$

$(a+b)^{1}=a+b\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 2\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1\ \ \ 1$

$(a+b)^{2}=a^{2}+2ab+b^{2}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 3\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1\ \ \ \ 2\ \ \ \ 1$

$(a+b)^{3}=a^{3}+3a^{2}b+3ab^{2}+b^{3}\ \ \ \ \ \ \ \ 4\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1\ \ \ \ 3\ \ \ \ 3\ \ \ \ 1$

(1) Can you write the results of $(a+b)^{4}$, $(a+b)^{5}$ based on the pattern in the table above?

$(a+b)^{4}=$______；

$(a+b)^{5}=$______.

(2) Please use the pattern in the table above to calculate the result of the following formula:

$2^{4}+4×2^{3}×(-\frac{1}{3})+6×2^{2}×(-\frac{1}{3})^{2}+4×2×(-\frac{1}{3})^{3}+(-\frac{1}{3})^{4}$.

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