fix(docs): fix grammar and remove duplicate words in documentation and source

Merge https://github.com/google/adk-python/pull/4335

Fixed 'the the' typos and grammatical errors in README, docstrings, and sample agents.

**Please ensure you have read the [contribution guide](https://github.com/google/adk-python/blob/main/CONTRIBUTING.md) before creating a pull request.**

### Link to Issue or Description of Change

**1. Link to an existing issue (if applicable):**

- Closes: N/A
- Related: N/A

**2. Or, if no issue exists, describe the change:**

**Problem:**
While exploring the repository, I noticed a few minor writing errors that impact professional readability:
1.  **Grammar:** A subject-verb agreement error in `README.md` and `llms.txt` ("guide the agents works together").
2.  **Duplicate Words:** Several instances of stuttering typos (repeating "the the" or "but the the") in docstrings within `src/` and the `contributing/samples/` directories.

**Solution:**
I have applied the following fixes to improve documentation quality:
*   Corrected phrasing to "guide the agents to work together" in the README.
*   Removed redundant instances of "the" in `eval_metrics.py`, `spanner/settings.py`, and the sample agent docstrings.
*   **No functional code or logic was altered.**

### Testing Plan

**Unit Tests:**

- [x] All unit tests pass locally.

_Summary:_
Since this PR is strictly limited to documentation, comments, and docstrings, no new tests were required. I ran the standard test suite to ensure no syntax errors were accidentally introduced, and everything passed successfully.

**Manual End-to-End (E2E) Tests:**
N/A — This is a static documentation fix.

### Checklist

- [x] I have read the [CONTRIBUTING.md](https://github.com/google/adk-python/blob/main/CONTRIBUTING.md) document.
- [x] I have performed a self-review of my own code.
- [ ] I have commented my code, particularly in hard-to-understand areas. (N/A)
- [ ] I have added tests that prove my fix is effective or that my feature works. (N/A - Doc fix only)
- [x] New and existing unit tests pass locally with my changes.
- [ ] I have manually tested my changes end-to-end. (N/A)
- [x] Any dependent changes have been merged and published in downstream modules.

### Additional context

_None._

COPYBARA_INTEGRATE_REVIEW=https://github.com/google/adk-python/pull/4335 from 88448844:fix-typos-and-docs 200668dad0c216b81d7008d9d6625ac135c9006e
PiperOrigin-RevId: 865530036

README.md

@@ -115,7 +115,7 @@ root_agent = Agent(
 
 ### Define a multi-agent system:
 
-Define a multi-agent system with coordinator agent, greeter agent, and task execution agent. Then ADK engine and the model will guide the agents works together to accomplish the task.
+Define a multi-agent system with coordinator agent, greeter agent, and task execution agent. Then ADK engine and the model will guide the agents to work together to accomplish the task.
 
 ```python
 from google.adk.agents import LlmAgent, BaseAgent

contributing/samples/adk_issue_formatting_agent/agent.py

@@ -96,7 +96,7 @@ def add_comment_to_issue(issue_number: int, comment: str) -> dict[str, any]:
     comment: comment to add
 
   Returns:
-    The the status of this request, with the applied comment when successful.
+    The status of this request, with the applied comment when successful.
   """
   print(f"Attempting to add comment '{comment}' to issue #{issue_number}")
   url = f"{GITHUB_BASE_URL}/repos/{OWNER}/{REPO}/issues/{issue_number}/comments"
@@ -119,7 +119,7 @@ def list_comments_on_issue(issue_number: int) -> dict[str, any]:
     issue_number: issue number of the GitHub issue
 
   Returns:
-    The the status of this request, with the list of comments when successful.
+    The status of this request, with the list of comments when successful.
   """
   print(f"Attempting to list comments on issue #{issue_number}")
   url = f"{GITHUB_BASE_URL}/repos/{OWNER}/{REPO}/issues/{issue_number}/comments"

contributing/samples/adk_pr_triaging_agent/agent.py

@@ -166,7 +166,7 @@ def add_label_to_pr(pr_number: int, label: str) -> dict[str, Any]:
       label: the label to add
 
   Returns:
-      The the status of this request, with the applied label and response when
+      The status of this request, with the applied label and response when
       successful.
   """
   print(f"Attempting to add label '{label}' to PR #{pr_number}")
@@ -201,7 +201,7 @@ def add_comment_to_pr(pr_number: int, comment: str) -> dict[str, Any]:
     comment: the comment to add
 
   Returns:
-    The the status of this request, with the applied comment when successful.
+    The status of this request, with the applied comment when successful.
   """
   print(f"Attempting to add comment '{comment}' to issue #{pr_number}")
 

contributing/samples/adk_triaging_agent/agent.py

@@ -230,7 +230,7 @@ def change_issue_type(issue_number: int, issue_type: str) -> dict[str, Any]:
     issue_type: issue type to assign
 
   Returns:
-    The the status of this request, with the applied issue type when successful.
+    The status of this request, with the applied issue type when successful.
   """
   print(
       f"Attempting to change issue type '{issue_type}' to issue #{issue_number}"

llms-full.txt

@@ -101,7 +101,7 @@ root_agent = Agent(
 
 ### Define a multi-agent system:
 
-Define a multi-agent system with coordinator agent, greeter agent, and task execution agent. Then ADK engine and the model will guide the agents works together to accomplish the task.
+Define a multi-agent system with coordinator agent, greeter agent, and task execution agent. Then ADK engine and the model will guide the agents to work together to accomplish the task.
 
 ```python
 from google.adk.agents import LlmAgent, BaseAgent

llms.txt

@@ -103,7 +103,7 @@ root_agent = Agent(
 Define a multi-agent system:
 
 
-Define a multi-agent system with coordinator agent, greeter agent, and task execution agent. Then ADK engine and the model will guide the agents works together to accomplish the task.
+Define a multi-agent system with coordinator agent, greeter agent, and task execution agent. Then ADK engine and the model will guide the agents to work together to accomplish the task.
 
 
 from google.adk.agents import LlmAgent, BaseAgent

src/google/adk/evaluation/eval_metrics.py

@@ -189,7 +189,7 @@ class ToolTrajectoryCriterion(BaseCriterion):
       Actual tool calls: [T1, T1.1, T2, T2.1, T2.2, T3, T3.1]
 
       While the tool calls T1, T2 and T3 happened in the "Actual" and in
-      the same order as "Expected", but the the tool calls T4 is missing.
+      the same order as "Expected", but the tool calls T4 is missing.
     """
 
     ANY_ORDER = 2
@@ -215,7 +215,7 @@ class ToolTrajectoryCriterion(BaseCriterion):
       Actual tool calls: [T1, T1.1, T2, T2.1, T2.2, T3, T3.1]
 
       While the tool calls T1, T2 and T3 happened in the "Actual" and in
-      the same order as "Expected", but the the tool calls T4 is missing.
+      the same order as "Expected", but the tool calls T4 is missing.
     """
 
   match_type: MatchType = Field(

src/google/adk/tools/spanner/settings.py

@@ -150,7 +150,7 @@ class SpannerVectorStoreSettings(BaseModel):
   """Required. The name of the embedding column to search in the vector store table."""
 
   vector_length: int
-  """Required. The the dimension of the vectors in the `embedding_column`."""
+  """Required. The dimension of the vectors in the `embedding_column`."""
 
   vertex_ai_embedding_model_name: str
   """Required. The Vertex AI embedding model name, which is used to generate embeddings for vector store and vector similarity search.