
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
Given $$(x+1)^{10}=a_{1}+a_{2}x+a_{3}x^{2}+\ldots+a_{11}x^{10}.$$ If the sequence $a_1, a_2, a_3, \ldots, a_k$ ($1\leq k\leq 11$, $k\in \mathbb{Z}$) is a monotonically increasing sequence, then the maximum value of $k$ is \_\_\_\_\_\_.

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