
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 the set \( M = \{1, 2, \cdots, 19\} \) and \( A = \{a_1, a_2, \cdots, a_k\} \subseteq M \). Determine the smallest \( k \), such that for any \( b \in M \), there exist \( a_i, a_j \in A \), satisfying \( a_i = b \) or \( a_i \pm a_j = b\) (where \(a_i\) and \(a_j\) can be the same).

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