
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 a function $f(x)$ and a real number $M$, if there exist $m$ and $n \in \mathbb{N}^+$ such that $f(m) + f(m+1) + f(m+2) + \ldots + f(m+n) = M$, then $(m, n)$ is called a "growth point" of the function $f(x)$ with respect to $M$. If $(1, 2)$ is a growth point of the function $f(x) = \cos(\frac{\pi}{2}x + \frac{\pi}{3})$ with respect to $M$, find the value of $M$. If $f(x) = 2x + 1$ and $M = 105$, find the number of growth points of the function $f(x)$ with respect to $M$.

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