
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
Expanding  $(1+0.2)^{1000}$  by the binomial theorem and doing no further manipulation gives \begin{eqnarray*} &\ & \binom{1000}{0}(0.2)^0+\binom{1000}{1}(0.2)^1+\binom{1000}{2}(0.2)^2+\cdots+\binom{1000}{1000}(0.2)^{1000} &\ & = A_0 + A_1 + A_2 + \cdots + A_{1000}, \end{eqnarray*} where  $A_k = \binom{1000}{k}(0.2)^k$  for  $k = 0,1,2,\ldots,1000$ . For which  $k$  is  $A_k$  the largest?

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