
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
Consider the sequence $\left\{a_{n}\right\}$ which satisfies:
\[ a_{1}=\frac{1}{4}, \, a_{n+1}=a_{n}+a_{n}^{2} \quad (n \in \mathbf{Z}_{+}) . \]

Let \( T_{2020}=\frac{1}{a_{1}+1}+\frac{1}{a_{2}+1}+\cdots+\frac{1}{a_{2020}+1} \).

If the value of $T_{2020}$ is within the interval $(k, k+1)$, determine the integer $k$.

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