
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 the arithmetic sequences $\{a\_n\}$ and $\{b\_n\}$ both with a common difference of $1$, their first terms are $a\_1$ and $b\_1$ respectively, and $a\_1 + b\_1 = 5$, $a\_1$, $b\_1 \in \mathbb{N}^+$. Let $c\_n = a\_{b\_n}$ ($n \in \mathbb{N}^+$), calculate the sum of the first $10$ terms of the sequence $\{c\_n\}$.

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