
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

Let $\left\{a_{n}\right\}$ be a sequence of positive integers that is monotonically increasing, i.e.,

$$
1 \leqq a_{1} \leqq a_{2} \leqq \ldots \leqq a_{n} \leqq \ldots
$$

For any integer $m \geqq 1$, let $b_{m}$ be the index of the first element in the sequence $\left\{a_{n}\right\}$ that is at least $m$.

If $a_{19}=88$, then within what range does the sum

$$
a_{1}+a_{2}+\ldots+a_{19}+b_{1}+b_{2}+\ldots+b_{88}
$$

lie?

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