
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 circle M: $(x+\cos q)^2 + (y-\sin q)^2 = 1$ and a line $l: y = kx$, consider the following four propositions:

① For any real numbers $k$ and $q$, the line $l$ is tangent to the circle M.

② For any real numbers $k$ and $q$, the line $l$ and the circle M intersect.

③ For any real number $q$, there exists a real number $k$ such that the line $l$ is tangent to the circle M.

④ For any real number $k$, there exists a real number $q$ such that the line $l$ is tangent to the circle M.

The valid propositions are coded __________ (write the codes of all valid propositions).

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